Divergence of Many-Body Perturbation Theory for Noncovalent Interactions of Large Molecules.

Prompted by recent reports of large errors in noncovalent interaction (NI) energies obtained from many-body perturbation theory (MBPT), we compare the performance of second-order MoÌ·ller-Plesset MBPT (MP2), spin-scaled MP2, dispersion-corrected semilocal density functional approximations (DFAs), and post-Kohn-Sham random phase approximation (RPA) for predicting binding energies of supramolecular complexes contained in the S66, L7, and S30L benchmarks. All binding energies are extrapolated to the basis set limit, corrected for basis set superposition errors, and compared to reference results of the domain-based local pair-natural orbital coupled-cluster (DLPNO-CCSD(T)) or better quality. Our results confirm that MP2 severely overestimates binding energies of large complexes, producing relative errors of over 100% for several benchmark compounds. RPA relative errors consistently range between 5 and 10%, significantly less than reported previously using smaller basis sets, whereas spin-scaled MP2 methods show limitations similar to MP2, albeit less pronounced, and empirically dispersion-corrected DFAs perform almost as well as RPA. Regression analysis reveals a systematic increase of relative MP2 binding energy errors with the system size at a rate of approximately 0.1% per valence electron, whereas the RPA and dispersion-corrected DFA relative errors are virtually independent of the system size. These observations are corroborated by a comparison of computed rotational constants of organic molecules to gas-phase spectroscopy data contained in the ROT34 benchmark. To analyze these results, an asymptotic adiabatic connection symmetry-adapted perturbation theory (AC-SAPT) is developed, which uses monomers at full coupling, whose ground-state density is constrained to the ground-state density of the complex. Using the fluctuation-dissipation theorem, we obtain a nonperturbative "screened second-order" expression for the dispersion energy in terms of monomer quantities, which is exact for non-overlapping subsystems and free of induction terms; a first-order RPA-like approximation to the Hartree, exchange, and correlation kernel recovers the macroscopic Lifshitz limit. The AC-SAPT expansion of the interaction energy is obtained from Taylor expansion of the coupling strength integrand. Explicit expressions for the convergence radius of the AC-SAPT series are derived within RPA and MBPT and numerically evaluated. While the AC-SAPT expansion is always convergent for nondegenerate monomers when RPA is used, it is found to spuriously diverge for second-order MBPT, except for the smallest and least polarizable monomers. The divergence of the AC-SAPT series for MBPT is numerically confirmed within RPA; prior numerical results on the convergence of the SAPT expansion for MBPT methods are revisited and support this conclusion once sufficiently high orders are included. The cause of the failure of MBPT methods for NIs of large systems is missing or incomplete "electrodynamic" screening of the Coulomb interaction due to induced particle-hole pairs between electrons in different monomers, leaving the effective interaction too strong for AC-SAPT to converge. Hence, MBPT cannot be considered reliable for quantitative predictions of NIs, even in moderately polarizable molecules with a few tens of atoms. The failure to accurately account for electrodynamic polarization makes MBPT qualitatively unsuitable for applications such as NIs of nanostructures, macromolecules, and soft materials; more robust nonperturbative approaches such as RPA or coupled cluster methods should be used instead whenever possible.

Structure Elucidation of a Melam-Melem Adduct by a Combined Approach of Synchrotron X-ray Diffraction and DFT Calculations.

Melam-melem (1:1), an adduct compound that can be obtained from dicyandiamide in autoclave reactions at 450 °C and elevated ammonia pressure, had previously been described based on mass spectrometry and NMR spectroscopy, but only incompletely characterized. The crystal structure of this compound has now been elucidated by means of synchrotron microfocus diffraction and subsequent quantum-chemical structure optimization applying DFT methods. The structure was refined in triclinic space group P 1 ‾ based on X-ray data. Cell parameters of a=4.56(2), b=19.34(8), c=21.58(11) Å, α=73.34(11)°, ß=89.1(2)°, and γ=88.4(2)° were experimentally obtained. The resulting cell volumes agree with the DFT optimized value to within 7 %. Molecular units in the structure form stacks that are interconnected by a vast array of hydrogen bridge interactions. Remarkably large melam dihedral angles of 48.4° were found that allow melam to interact with melem molecules from different stack layers, thus forming a 3D network. π-stacking interactions appear to play no major role in this structure.

Robust Periodic Fock Exchange with Atom-Centered Gaussian Basis Sets.

In this work, we present a robust implementation of the periodic Fock exchange for atom-centered Gaussian-type orbitals (GTOs). We discuss the divergence, appearing in the formulation of the periodic Fock exchange in the case of a finite number of k-points, and compare two schemes that remove it. These are the minimum image convention (MIC) and the truncated Coulomb interaction (TCI) that we use here in combination with k-meshes. We observe artifacts in four-center integrals of GTOs, when evaluated in the TCI scheme. They carry over to the exchange and density matrices of Hartree-Fock calculations for TCI but are absent in MIC. At semiconducting and insulating systems, we show that both MIC and TCI yield the same energies for a sufficiently large supercell or k-mesh, but the self-consistent field algorithm is more stable for MIC. We therefore conclude that the MIC is superior to TCI and validate our implementation by comparing not only to other GTO-based calculations but also by demonstrating excellent agreement with results of plane-wave codes for sufficiently large Gaussian basis sets.

Topochemical conversion of an imine- into a thiazole-linked covalent organic framework enabling real structure analysis.

Stabilization of covalent organic frameworks (COFs) by post-synthetic locking strategies is a powerful tool to push the limits of COF utilization, which are imposed by the reversible COF linkage. Here we introduce a sulfur-assisted chemical conversion of a two-dimensional imine-linked COF into a thiazole-linked COF, with full retention of crystallinity and porosity. This post-synthetic modification entails significantly enhanced chemical and electron beam stability, enabling investigation of the real framework structure at a high level of detail. An in-depth study by electron diffraction and transmission electron microscopy reveals a myriad of previously unknown or unverified structural features such as grain boundaries and edge dislocations, which are likely generic to the in-plane structure of 2D COFs. The visualization of such real structural features is key to understand, design and control structure-property relationships in COFs, which can have major implications for adsorption, catalytic, and transport properties of such crystalline porous polymers.

Random Phase Approximation for Periodic Systems Employing Direct Coulomb Lattice Summation.

A method to compute ground state correlation energies from the random phase approximation (RPA) is presented for molecular and periodic systems on an equal footing. The supermatrix representation of the Hartree kernel in canonical orbitals is translation-symmetry adapted and factorized by the resolution of the identity (RI) approximation. Orbital expansion and RI factorization employ atom-centered Gaussian-type basis functions. Long ranging Coulomb lattice sums are evaluated in direct space with a revised recursive multipole method that works also for irreducible representations different from Γ. The computational cost of this RI-RPA method scales as [Formula: see text](N4) with the system size in direct space, N, and as [Formula: see text](Nk2) with the number of sampled k-points in reciprocal space, Nk. For chain and film models, the exploration of translation symmetry with 10 k-points along each periodic direction reduces the computational cost by a factor of around 10-100 compared to equivalent Γ-point supercell calculations.

Density Functional Theory for Molecular and Periodic Systems Using Density Fitting and Continuous Fast Multipole Methods.

An implementation of Kohn-Sham density functional theory within the TURBOMOLE program package with Gaussian-type orbitals (GTO) as basis functions is reported that treats molecular and periodic systems of any dimensionality on an equal footing. Its key component is a combination of density fitting/resolution of identity (DF) approximation and continuous fast multipole method (CFMM) applied for the electronic Coulomb term. This DF-CFMM scheme operates entirely in the direct space and partitions Coulomb interactions into far-field part evaluated using multipole expansions and near-field contribution calculated employing density fitting. Computational efficiency and favorable scaling behavior of our implementation approaching O(N) for the formation of Kohn-Sham matrix is demonstrated for various molecular and periodic systems including three-dimensional models with unit cells containing up to 640 atoms and 19072 GTO basis functions.

Analytical First-Order Molecular Properties and Forces within the Adiabatic Connection Random Phase Approximation.

The random phase approximation (RPA) is an increasingly popular method for computing molecular ground-state correlation energies within the adiabatic connection fluctuation-dissipation theorem framework of density functional theory. We present an efficient analytical implementation of first-order RPA molecular properties and nuclear forces using the resolution-of-the-identity (RI) approximation and imaginary frequency integration. The centerpiece of our approach is a variational RPA energy Lagrangian invariant under unitary transformations of occupied and virtual reference orbitals, respectively. Its construction requires the solution of a single coupled-perturbed Kohn-Sham equation independent of the number of perturbations. Energy gradients with respect to nuclear displacements and other first-order properties such as one-particle densities or dipole moments are obtained from partial derivatives of the Lagrangian. Our RPA energy gradient implementation exhibits the same [Formula: see text] scaling with system size N as a single-point RPA energy calculation. In typical applications, the cost for computing the entire gradient vector with respect to nuclear displacements is ∼5 times that of a single-point RPA energy calculation. Derivatives of the quadrature nodes and weights used for frequency integration are essential for RPA gradients with an accuracy consistent with RPA energies and can be included in our approach. The quality of RPA equilibrium structures is assessed by comparison to accurate theoretical and experimental data for covalent main group compounds, weakly bonded dimers, and transition metal complexes. RPA outperforms semilocal functionals as well as second-order Møller-Plesset (MP2) theory, which fails badly for the transition metal compounds. Dipole moments of polarizable molecules and weakly bound dimers show a similar trend. RPA harmonic vibrational frequencies are nearly of coupled cluster singles, doubles, and perturbative triples quality for a set of main group compounds. Compared to the ring-coupled cluster based implementation of Rekkedal et al. [J. Chem. Phys. 2013, 139, 081101.], our method scales better by two powers of N and supports a semilocal Kohn-Sham reference. The latter is essential for the good performance of RPA in small-gap systems.

Structures and vibrational spectroscopy of partially reduced gas-phase cerium oxide clusters.

This work demonstrates that the most stable structures of even small gas-phase aggregates of cerium oxide with 2-5 cerium atoms show structural motifs reminiscent of the bulk ceria. This is different from main group and transition metal oxide clusters, which often display structural features that are distinctly different from the bulk structure. The structures of Ce(2)O(2)(+), Ce(3)O(4)(+), and (CeO(2))(m)CeO(+) clusters (m = 0-4) are unambiguously determined by a combination of global structure optimizations at the density functional theory level and infrared vibrational predissociation spectroscopy of the cluster-rare gas atom complexes. The structures of Ce(2)O(2)(+) and Ce(2)O(3)(+) exhibit a Ce-O-Ce-O four-membered ring with characteristic absorptions between 430 and 680 cm(-1). Larger clusters have common structural features containing fused Ce-O-Ce-O four-membered rings which lead to intense absorption bands at around 500 and 650 cm(-1). Clusters containing a terminal Ce=O bond show a characteristic absorption band between 800 and 840 cm(-1). For some cluster sizes multiple isomers are observed. Their individual infrared signatures are identified by tuning their relative population through the choice of He, Ne or Ar messenger atoms. The present results allow us to benchmark different density functionals which yield different degrees of localization of unpaired electrons in Ce 4f states.

Electron localization in defective ceria films: a study with scanning-tunneling microscopy and density-functional theory.

Scanning-tunneling microscopy and density-functional theory have been employed to identify the spatial correlation between an oxygen vacancy and the associated Ce(3+) ion pair in a defective CeO(2)(111) film. The two Ce(3+) ions can occupy different cationic shells around the vacancy. The resulting variation in the chemical environment leads to a splitting of the filled Ce(3+) f levels, which is detected with STM spectroscopy. The position of the Ce(3+) ion pair is reflected in characteristic defect patterns observed in empty-state STM images, which arise from the bright appearance of Ce(4+) ions next to the defect while the Ce(3+) remain dark. Both findings demonstrate that at least one excess electron localizes in a Ce ion that is not adjacent to the O vacancy.

Linear Scaling Hierarchical Integration Scheme for the Exchange-Correlation Term in Molecular and Periodic Systems.

An adaptive numerical integration scheme for efficient evaluation of the exchange-correlation term using localized basis functions and atom-centered grids is presented. The method treats molecules and systems with periodic boundary conditions on an equal footing. Its computational efficiency and O(N) scaling with the system size is achieved by a hierarchical spatial grouping of basis functions and grid points using an octree. This allows for an efficient screening of negligible contributions and an optimal use of hardware-optimized matrix-matrix multiplication subroutines, such as BLAS. The implementation of the method within the TURBOMOLE program package demonstrates consistent accuracy and efficiency across molecular and periodic systems.

Modeling environmental effects on charge density distributions in polar organometallics: validation of embedded cluster models for the methyl lithium crystal.

The charge density and its Laplacian at the Li-C and C-H bond critical points and other features of the electron density distribution of the methyl lithium crystal have been compared by density functional methods for (i) the isolated (LiCH(3))(4) tetramer or larger clusters, (ii) for quantum mechanically treated clusters in polarizable continuum model (PCM) surroundings, (iii) for clusters augmented by the periodic electrostatic embedded cluster model (PEECM), and for (iv) the periodic crystal. Comparisons with identical functional and basis sets indicate that both PCM and PEECM embedding of only a tetramer did not fully account for the environmental effect. In contrast, embedding of a full unit cell gave results that were essentially converged to the periodic crystal data. Effects of basis set and exchange correlation functional on the QTAIM bond descriptors are of a comparable order of magnitude as the crystal environmental effects. In this context, embedded cluster computations provide distinct advantages over explicit solid-state calculations with respect to their freedom of the choice of computational and theoretical level. This is demonstrated by embedded MP2 calculations.

Resolution of identity approximation for the Coulomb term in molecular and periodic systems.

A new formulation of resolution of identity approximation for the Coulomb term is presented, which uses atom-centered basis and auxiliary basis functions and treats molecular and periodic systems of any dimensionality on an equal footing. It relies on the decomposition of an auxiliary charge density into charged and chargeless components. Applying the Coulomb metric under periodic boundary conditions constrains the explicit form of the charged part. The chargeless component is determined variationally and converged Coulomb lattice sums needed for its determination are obtained using chargeless linear combinations of auxiliary basis functions. The lattice sums are partitioned in near- and far-field portions which are treated through an analytical integration scheme employing two- and three-center electron repulsion integrals and multipole expansions, respectively, operating exclusively in real space. Our preliminary implementation within the TURBOMOLE program package demonstrates consistent accuracy of the method across molecular and periodic systems. Using common auxiliary basis sets the errors of the approximation are small, in average about 20 muhartree per atom, for both molecular and periodic systems.

Point defects in CaF2 and CeO2 investigated by the periodic electrostatic embedded cluster method.

A periodic electrostatic embedding scheme is presented that uses the periodic fast multipole method. The convergence of properties with increasing cluster size is examined for cluster models of calcium fluoride. Properties investigated are the electron density, the density of states, the electronic excitation of color centers, and energies of defect formation. The embedded cluster method is applied to CeO(2) and oxygen vacancies in bulk CeO(2) as well as on its (111) surface. Employing the PBE0 functional, vacancy formation energies of 3.0 and 3.3 eV have been obtained for the bulk and the (111) surface, respectively. Formation of subsurface defects requires 3.33 eV (singlet open shell). The localization of the electrons left behind on defect formation in Ce 4f states is discussed. Occupied Ce 4f states are well localized on nearest Ce atoms for surface and subsurface vacancies. Localization apart from the vacancy was obtained for bulk. The total CPU time spent on the embedding part did not exceed 30 s on a single CPU even if 8000 basis functions of the cluster are involved.