Contributions beyond direct random-phase approximation in the binding energy of solid ethane, ethylene, and acetylene.

The random-phase approximation (RPA) includes a subset of higher than second-order correlation-energy contributions, but stays in the same complexity class as the second-order Møller-Plesset perturbation theory (MP2) in both Gaussian-orbital and plane-wave codes. This makes RPA a promising ab initio electronic structure approach for the binding energies of molecular crystals. Still, some issues stand out in practical applications of RPA. Notably, compact clusters of nonpolar molecules are poorly described, and the interaction energies strongly depend on the reference single-determinant state. Using the many-body expansion of the binding energy of a crystal, we investigate those issues and the effect of beyond-RPA corrections. We find the beneficial effect of quartic-scaling exchange and non-ring coupled-cluster doubles corrections. The nonadditive interactions in compact trimers of molecules are improved by using the self-consistent Hartree-Fock orbitals instead of the usual Kohn-Sham states, but this kind of orbital input also leads to underestimated dimer energies. Overall, a substantial improvement over the RPA with a renormalized singles approach is possible at a modest quartic-scaling cost, which encourages further research into additional RPA corrections.

Post-Kohn-Sham Random-Phase Approximation and Correction Terms in the Expectation-Value Coupled-Cluster Formulation.

Using expectation-value coupled-cluster theory and many-body perturbation theory (MBPT), we formulate a series of corrections to the post-Kohn-Sham (post-KS) random-phase approximation (RPA) energy. The beyond-RPA terms are of two types: those accounting for the non-Hartree-Fock reference and those introducing the coupled-cluster doubles non-ring contractions. The contributions of the former type, introduced via the semicanonical orbital basis, drastically reduce the binding strength in noncovalent systems. The good accuracy is recovered by the attractive third-order doubles correction referred to as Ec2g. The existing RPA approaches based on KS orbitals neglect most of the proposed corrections but can perform well thanks to error cancellation. The proposed method accounts for every contribution in the state-of-the-art renormalized second-order perturbation theory (rPT2) approach but adds additional terms which initially contribute in the third order of MBPT. The cost of energy evaluation scales as noniterative O(N4) in the implementation with low-rank tensor decomposition. The numerical tests of the proposed approach demonstrate accurate results for noncovalent dimers of polar molecules and for the challenging many-body noncovalent cluster of CH4···(H2O)20.

Efficient Calculation of the Dispersion Energy for Multireference Systems with Cholesky Decomposition: Application to Excited-State Interactions.

Accurate and efficient prediction of dispersion interactions in excited-state complexes poses a challenge due to the complex nature of electron correlation effects that need to be simultaneously considered. We propose an algorithm for computing the dispersion energy in nondegenerate ground- or excited-state complexes with arbitrary spin. The algorithm scales with the fifth power of the system size due to employing Cholesky decomposition of Coulomb integrals and a recently developed recursive formula for density response functions of the monomers. As a numerical illustration, we apply the new algorithm in the framework of multiconfigurational symmetry adapted perturbation theory, SAPT(MC), to study interactions in dimers with localized excitons. The SAPT(MC) analysis reveals that the dispersion energy may be the main force stabilizing excited-state dimers.

Assessment of random phase approximation and second-order Møller-Plesset perturbation theory for many-body interactions in solid ethane, ethylene, and acetylene.

The relative energies of different phases or polymorphs of molecular solids can be small, less than a kilojoule/mol. A reliable description of such energy differences requires high-quality treatment of electron correlations, typically beyond that achievable by routinely applicable density functional theory (DFT) approximations. At the same time, high-level wave function theory is currently too computationally expensive. Methods employing an intermediate level of approximations, such as Møller-Plesset (MP) perturbation theory and the random phase approximation (RPA), are potentially useful. However, their development and application for molecular solids has been impeded by the scarcity of necessary benchmark data for these systems. In this work, we employ the coupled-cluster method with singles, doubles, and perturbative triples to obtain a reference-quality many-body expansion of the binding energy of four crystalline hydrocarbons with a varying π-electron character: ethane, ethene, and cubic and orthorhombic forms of acetylene. The binding energy is resolved into explicit dimer, trimer, and tetramer contributions, which facilitates the analysis of errors in the approximate approaches. With the newly generated benchmark data, we test the accuracy of MP2 and non-self-consistent RPA. We find that both of the methods poorly describe the non-additive many-body interactions in closely packed clusters. Using different DFT input states for RPA leads to similar total binding energies, but the many-body components strongly depend on the choice of the exchange-correlation functional.

Efficient Adiabatic Connection Approach for Strongly Correlated Systems: Application to Singlet-Triplet Gaps of Biradicals.

Strong electron correlation can be captured with multireference wave function methods, but an accurate description of the electronic structure requires accounting for the dynamic correlation, which they miss. In this work, a new approach for the correlation energy based on the adiabatic connection (AC) is proposed. The ACn method accounts for terms up to order n in the coupling constant, and it is size-consistent and free from instabilities. It employs the multireference random phase approximation and the Cholesky decomposition technique, leading to a computational cost growing with the fifth power of the system size. Because of the dependence on only one- and two-electron reduced density matrices, ACn is more efficient than existing ab initio multireference dynamic correlation methods. ACn affords excellent results for singlet-triplet gaps of challenging organic biradicals. The development presented in this work opens new perspectives for accurate calculations of systems with dozens of strongly correlated electrons.

Dataset of noncovalent intermolecular interaction energy curves for 24 small high-spin open-shell dimers.

We introduce a dataset of 24 interaction energy curves of open-shell noncovalent dimers, referred to as the O24 × 5 dataset. The dataset consists of high-spin dimers up to 11 atoms selected to assure diversity with respect to interaction types: dispersion, electrostatics, and induction. The benchmark interaction energies are obtained at the restricted open-shell CCSD(T) level of theory with complete basis set extrapolation (from aug-cc-pVQZ to aug-cc-pV5Z). We have analyzed the performance of selected wave function methods MP2, CCSD, and CCSD(T) as well as the F12a and F12b variants of coupled-cluster theory. In addition, we have tested dispersion-corrected density functional theory methods based on the PBE exchange-correlation model. The O24 × 5 dataset is a challenge to approximate methods due to the wide range of interaction energy strengths it spans. For the dispersion-dominated and mixed-type subsets, any tested method that does not include the triples contribution yields errors on the order of tens of percent. The electrostatic subset is less demanding with errors that are typically an order of magnitude smaller than the mixed and dispersion-dominated subsets.

Random-Phase Approximation in Many-Body Noncovalent Systems: Methane in a Dodecahedral Water Cage.

The many-body expansion (MBE) of energies of molecular clusters or solids offers a way to detect and analyze errors of theoretical methods that could go unnoticed if only the total energy of the system was considered. In this regard, the interaction between the methane molecule and its enclosing dodecahedral water cage, CH4···(H2O)20, is a stringent test for approximate methods, including density functional theory (DFT) approximations. Hybrid and semilocal DFT approximations behave erratically for this system, with three- and four-body nonadditive terms having neither the correct sign nor magnitude. Here, we analyze to what extent these qualitative errors in different MBE contributions are conveyed to post-Kohn-Sham random-phase approximation (RPA), which uses approximate Kohn-Sham orbitals as its input. The results reveal a correlation between the quality of the DFT input states and the RPA results. Moreover, the renormalized singles energy (RSE) corrections play a crucial role in all orders of the many-body expansion. For dimers, RSE corrects the RPA underbinding for every tested Kohn-Sham model: generalized-gradient approximation (GGA), meta-GGA, (meta-)GGA hybrids, as well as the optimized effective potential at the correlated level. Remarkably, the inclusion of singles in RPA can also correct the wrong signs of three- and four-body nonadditive energies as well as mitigate the excessive higher-order contributions to the many-body expansion. The RPA errors are dominated by the contributions of compact clusters. As a workable method for large systems, we propose to replace those compact contributions with CCSD(T) energies and to sum up the remaining many-body contributions up to infinity with supermolecular or periodic RPA. As a demonstration of this approach, we show that for RPA(PBE0)+RSE it suffices to apply CCSD(T) to dimers and 30 compact, hydrogen-bonded trimers to get the methane-water cage interaction energy to within 1.6% of the reference value.

Assessment of SAPT(DFT) with meta-GGA functionals.

This work examines the suitability of meta-GGA functionals for symmetry-adapted perturbation theory (SAPT) calculations. The assessment is based on the term-by-term comparison with the benchmark SAPT variant based on coupled-cluster singles and doubles description of monomers, SAPT(CCSD). Testing systems include molecular complexes ranging from strong to weak and the He dimer. The following nonempirical meta-GGAs are examined: TPSS, revTPSS, MVS, SCAN, and SCAN0 with and without the asymptotic correction (AC) of the exchange-correlation potential. One range-separated meta-GGA functional, LC-PBETPSS, is also included. The AC-corrected pure meta-GGAs (with the exception of MVS) represent a definite progress in SAPT(DFT) compared to pure GGA, such as PBEAC, with their more consistent predictions of energy components. However, none of the meta-GGAs is better than the hybrid GGA approach SAPT(PBE0AC). The SAPT(DFT) electrostatic energy offers the most sensitive probe of the quality of the underlying DFT density. Both SCAN- and TPSS-based electrostatic energies agree with reference to within 5% or better which is an excellent result. We find that SCAN0 can be used in SAPT without the AC correction. The long-range corrected LC-PBETPSS is a reliable performer both for the components and total interaction energies.

Random Phase Approximation Applied to Many-Body Noncovalent Systems.

The random phase approximation (RPA) has received considerable interest in the field of modeling systems where noncovalent interactions are important. Its advantages over widely used density functional theory (DFT) approximations are the exact treatment of exchange and the description of long-range correlation. In this work, we address two open questions related to RPA. First, we demonstrate how accurately RPA describes nonadditive interactions encountered in many-body expansion of a binding energy. We consider three-body nonadditive energies in molecular and atomic clusters. Second, we address how the accuracy of RPA depends on input provided by different DFT models, without resorting to self-consistent RPA procedure, which is currently impractical for calculations employing periodic boundary conditions. We find that RPA based on the SCAN0 and PBE0 models, that is, hybrid DFT, achieves an overall accuracy between CCSD and MP3 on a data set of molecular trimers from Rezác et al. ( J. Chem. Theory. Comput. 2015 , 11 , 3065 ). Finally, many-body expansion for molecular clusters and solids often leads to a large number of small contributions that need to be calculated with high precision. We therefore present a cubic-scaling (or self-consistent field (SCF)-like) implementation of RPA in atomic basis set, which is designed for calculations with high numerical precision.

The nature of three-body interactions in DFT: Exchange and polarization effects.

We propose a physically motivated decomposition of density functional theory (DFT) 3-body nonadditive interaction energies into the exchange and density-deformation (polarization) components. The exchange component represents the effect of the Pauli exclusion in the wave function of the trimer and is found to be challenging for density functional approximations (DFAs). The remaining density-deformation nonadditivity is less dependent upon the DFAs. Numerical demonstration is carried out for rare gas atom trimers, Ar2-HX (X = F, Cl) complexes, and small hydrogen-bonded and van der Waals molecular systems. None of the tested semilocal, hybrid, and range-separated DFAs properly accounts for the nonadditive exchange in dispersion-bonded trimers. By contrast, for hydrogen-bonded systems, range-separated DFAs achieve a qualitative agreement to within 20% of the reference exchange energy. A reliable performance for all systems is obtained only when the monomers interact through the Hartree-Fock potential in the dispersion-free Pauli blockade scheme. Additionally, we identify the nonadditive second-order exchange-dispersion energy as an important but overlooked contribution in force-field-like dispersion corrections. Our results suggest that range-separated functionals do not include this component, although semilocal and global hybrid DFAs appear to imitate it in the short range.

Employing Range Separation on the meta-GGA Rung: New Functional Suitable for Both Covalent and Noncovalent Interactions.

We devise a scheme for converting an existing exchange functional into its range-separated hybrid variant. The underlying exchange hole of the Becke-Roussel type has the exact second-order expansion in the interelectron distance. The short-range part of the resulting range-separated exchange energy depends on the kinetic energy density and the Laplacian even if the base functional lacks the dependence on these variables. The most successful practical realization of the scheme, named LC-PBETPSS, combines the range-separated Perdew-Burke-Ernzerhof (PBE) exchange lifted to the hybrid meta-generalized gradient approximation rung and the Tao-Perdew-Staroverov-Scuseria (TPSS) correlation. The value of the range-separation parameter is estimated theoretically and confirmed by empirical optimization. The D3 dispersion correction is recommended for all energy computations employing the presented functional. Numerical tests show remarkably robust performance of the method for noncovalent interaction energies, barrier heights, main-group thermochemistry, and excitation energies.

Tuned range-separated hybrid functionals in the symmetry-adapted perturbation theory.

The aim of this study is to present a performance test of optimally tuned long-range corrected (LRC) functionals applied to the symmetry-adapted perturbation theory (SAPT). In the present variant, the second-order energy components are evaluated at the coupled level of theory. We demonstrate that the generalized Kohn-Sham (GKS) description of monomers with optimally tuned LRC functionals may be essential for the quality of SAPT interaction energy components. This is connected to the minimization of a many-electron self-interaction error and exemplified by two model systems: polyacetylenes of increasing length and stretching of He 3 (+). Next we provide a comparison of SAPT approaches based on Kohn-Sham and GKS description of the monomers. We show that LRC leads to results better or comparable with the hitherto prevailing asymptotically corrected functionals. Finally, we discuss the advantages and possible limitations of SAPT based on LRC functionals.

Transition properties from the Hermitian formulation of the coupled cluster polarization propagator.

Theory of one-electron transition density matrices has been formulated within the time-independent coupled cluster method for the polarization propagator [R. Moszynski, P. S. Zuchowski, and B. Jeziorski, Coll. Czech. Chem. Commun. 70, 1109 (2005)]. Working expressions have been obtained and implemented with the coupled cluster method limited to single, double, and linear triple excitations (CC3). Selected dipole and quadrupole transition probabilities of the alkali earth atoms, computed with the new transition density matrices are compared to the experimental data. Good agreement between theory and experiment is found. The results obtained with the new approach are of the same quality as the results obtained with the linear response coupled cluster theory. The one-electron density matrices for the ground state in the CC3 approximation have also been implemented. The dipole moments for a few representative diatomic molecules have been computed with several variants of the new approach, and the results are discussed to choose the approximation with the best balance between the accuracy and computational efficiency.

Density functional theory approach to gold-ligand interactions: separating true effects from artifacts.

Donor-acceptor interactions are notoriously difficult and unpredictable for conventional density functional theory (DFT) methodologies. This work presents a reliable computational treatment of gold-ligand interactions of the donor-acceptor type within DFT. These interactions require a proper account of the ionization potential of the electron donor and electron affinity of the electron acceptor. This is accomplished in the Generalized Kohn Sham framework that allows one to relate these properties to the frontier orbitals in DFT via the tuning of range-separated functionals. A donor and an acceptor typically require different tuning schemes. This poses a problem when the binding energies are calculated using the supermolecular method. A two-parameter tuning for the monomer properties ensures that a common functional, optimal for both the donor and the acceptor, is found. A reliable DFT approach for these interactions also takes into account the dispersion contribution. The approach is validated using the water dimer and the (HAuPH3)2 aurophilic complex. Binding energies are computed for Au4 interacting with the following ligands: SCN(-), benzenethiol, benzenethiolate anion, pyridine, and trimethylphosphine. The results agree for the right reasons with coupled-cluster reference values.

Range-Separated meta-GGA Functional Designed for Noncovalent Interactions.

The accuracy of applying density functional theory to noncovalent interactions is hindered by errors arising from low-density regions of interaction-induced change in the density gradient, error compensation between correlation and exchange functionals, and dispersion double counting. A new exchange-correlation functional designed for noncovalent interactions is proposed to address these problems. The functional consists of the range-separated PBEsol exchange considered in two variants, pure and hybrid, and the semilocal correlation functional of Modrzejewski et al. (J. Chem. Phys. 2012, 137, 204121) designed with the constraint satisfaction technique to smoothly connect with a dispersion term. Two variants of dispersion correction are appended to the correlation functional: the atom-atom pairwise additive DFT-D3 model and the density-dependent many-body dispersion with self-consistent screening (MBD-rsSCS). From these building blocks, a set of four functionals is created to systematically examine the role of pure versus hybrid exchange and the underlying models for dispersion. The new functional is extensively tested on benchmark sets with diverse nature and size. Truly outstanding performance is demonstrated for water clusters of varying size, ionic hydrogen bonds, and thermochemistry of isodesmic n-alkane fragmentation reactions. The merits of each component of the new functional are discussed.

Density-dependent onset of the long-range exchange: a key to donor-acceptor properties.

Quantum mechanical methods based on the density functional theory (DFT) offer a realistic possibility of first-principles design of organic donor-acceptor systems and engineered band gap materials. This promise is contingent upon the ability of DFT to predict one-particle states accurately. Unfortunately, approximate functionals fail to align the orbital energies with ionization potentials. We describe a new paradigm for achieving this alignment. In the proposed model, an average electron-exchange hole separation controls the onset of the orbital-dependent exchange in approximate range-separated functionals. The correct description of one-particle states is thus achieved without explicit electron removal or attachment. Extensive numerical tests show that the proposed method provides physically sound orbital gaps and leads to excellent predictions of charge-transfer excitations and other properties critically depending on the tail of the electron density.

A first-principles-based correlation functional for harmonious connection of short-range correlation and long-range dispersion.

We present a physically motivated correlation functional belonging to the meta-generalized gradient approximation (meta-GGA) rung, which can be supplemented with long-range dispersion corrections without introducing double-counting of correlation contributions. The functional is derived by the method of constraint satisfaction, starting from an analytical expression for a real-space spin-resolved correlation hole. The model contains a position-dependent function that controls the range of the interelectronic correlations described by the semilocal functional. With minimal empiricism, this function may be adjusted so that the correlation model blends with a specific dispersion correction describing long-range contributions. For a preliminary assessment, our functional has been combined with an atom-pairwise dispersion correction and full Hartree-Fock (HF)-like exchange. Despite the HF-exchange approximation, its predictions compare favorably with reference interaction energies in an extensive set of non-covalently bound dimers.

Dispersion-free component of non-covalent interaction via mutual polarization of fragment densities.

Comprehensive tests within a diverse set of noncovalently bonded systems are carried out to assess the performance of the recently-developed dispersion-free approach in the framework of density functional theory [L. Rajchel, P. Zuchowski, M. Szczesniak, and G. Chalasinski, Phys. Rev. Lett. 104, 163001 (2010)]. A numerical algorithm which cures the convergence problems of the previous implementation is presented.

A density functional theory approach to noncovalent interactions via interacting monomer densities.

A recently proposed "DFT + dispersion" treatment (Rajchel et al., Phys. Rev. Lett., 2010, 104, 163001) is described in detail and illustrated by more examples. The formalism derives the dispersion-free density functional theory (DFT) interaction energy and combines it with the dispersion energy from separate DFT calculations. It consists of the self-consistent polarization of DFT monomers restrained by the exclusion principle via the Pauli blockade technique. Within the monomers a complete exchange-correlation potential should be used, but between them only the exact exchange operates. The application to a wide range of molecular complexes from rare-gas dimers to hydrogen-bonds to π-electron interactions shows good agreement with benchmark values.