Long-range Correlation Effects Research Articles

Exact Sum-Rule Approach to Polarizability and Asymptotic van der Waals Functionals─Derivation of Exact Single-Particle Benchmarks

Using a sum-rule approach, we develop an exact theoretical framework for polarizability and asymptotic van der Waals correlation energy functionals of small isolated objects. The functionals require only monomer ground-state properties as input. Functional evaluation proceeds via solution of a single position-space differential equation, without the usual summations over excited states or frequency integrations. Explicit functional forms are reported for reference physical systems, including atomic hydrogen and single electrons subject to harmonic confinement, and immersed in a spherical-well potential. A direct comparison to the popular Vydrov-van Voorhis density functional shows that the best performance is obtained when density decay occurs at atomic scales. The adopted sum-rule approach implies general validity of our theory, enabling exact benchmarking of van der Waals density functionals and direct inspection of the subtle long-range correlation effects that constitute a major challenge for approximate (semi)local density functionals.

A shortcut to the thermodynamic limit for quantum many-body calculations of metals

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.

Halogen-Halogen Nonbonded Interactions.

Halogen–halogen nonbonded interactions were studied for methyl halides and phenyl halides using both B3LYP and MP2 along with 6-311+G* and aug-cc-pVTZ. With the methyl halides, the linear approach was found to lead to little stabilization, whereas the “90°” approach gave 1–2 kcal/mol. This modest stabilization was due to long-range electron correlation effects. The lowest-energy arrangement had the molecules side-by-side, with the major stabilization being derived from halogen–hydrogen interactions. The results for methyl bromide were quite similar. Chlorobenzene dimer with the 90° orientation gave a small stabilization energy, but the best arrangement had the two benzene rings oriented over each other. The meta orientation of the chlorines had a lower energy than ortho or para. The dimerization energy was larger than that for two benzene rings sitting directly above each other, suggesting that whereas Cl···Cl interaction is not very important, the effect of the halogen on the electron distribution does have an effect. This suggests that much of the crystallographic results for these compounds may not be due to halogen–halogen interactions but rather the interaction between the substituted benzene rings along with crystal forces.

Effects of masking titanium with a one-atom-thick carbon layer on the adsorption of nitrogen monoxide, nitrogen dioxide, ozone, and formaldehyde

We investigated the adsorption of each of four pollutant molecules, namely nitrogen monoxide, nitrogen dioxide, ozone, and formaldehyde, on the surface of a [0001] titanium slab, when it has adsorbed a one-atom-thick carbon layer. The tile is made of four titanium layers. Density functional theory and molecular dynamics were utilized. Atmospheric pressure and 300 K were considered. We found chemisorption of the four molecules on the [0001] titanium surface. Then, we optimized the interaction of the metal surface and a one-atom-thick carbon layer. The carbon atoms end up bound to titanium atoms on the surface. Carbon atoms' quasilinear chains are formed. The adsorption energy per carbon atom is − 2.054 eV. There is no migration of carbon atoms to the bulk. We found significant changes in the adsorption properties. The new system repels formaldehyde and nitrogen oxide molecules. In the ozone case, chemisorption of one oxygen atom on the surface is found, and the remaining oxygen molecule is repelled. In the nitrogen dioxide case, chemisorption of one oxygen atom on the surface is found too, and the leftover fraction is also repelled. These calculations were performed using GGA, and afterward, we utilized Vdw-DF2, which includes long-range correlation effects. The same results are obtained.

Molecule-surface interaction from van der Waals-corrected semilocal density functionals: The example of thiophene on transition-metal surfaces

Semi-local density functional approximations are widely used. None of them can capture the long-range van der Waals (vdW) attraction between separated subsystems, but they differ remarkably in the extent to which they capture intermediate-range vdW effects responsible for equilibrium bonds between neighboring small closed-shell subsystems. The local density approximation (LDA) often overestimates this effect, while the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) underestimates it. The strongly-constrained and appropriately normed (SCAN) meta-GGA often estimates it well. All of these semi-local functionals require an additive non-local correction such as the revised Vydrov-Van Voorhis 2010 (rVV10) to capture the long-range part. This work reports adsorption energies and the corresponding geometry of the aromatic thiophene (C$_4$H$_4$S) bound to transition metal surfaces. The adsorption process requires a genuine interplay of covalent and weak binding and requires a simultaneously accurate description of surface and adsorption energies with the correct prediction of the adsorption site. All these quantities must come from well balanced short and long-range correlation effects for a universally applicable method for weak interactions with chemical accuracy.

Generalized Valence Bond Perfect-Pairing Made Versatile Through Electron-Pairs Embedding.

We present an electron-pairs-based method employing a generalized valence bond perfect-pairing (GVB-PP) ansatz that provides a uniformly accurate description of systems where various types of electron correlation play a role and the GVB-PP wave function is a suitable reference. In the proposed EERPA-GVB approach, a GVB-PP energy is amended by adding correlation among electron pairs. The latter is achieved by embedding single pairs or couples of pairs in the environment of the other electron fragments and separately accounting for intra- and interfragment correlation effects. For this purpose, we employ truncated extended random phase approximation equations. Application of EERPA-GVB to systems governed by both short-range (energy barriers) and long-range (molecular interactions) correlation effects proves the good accuracy of the method. Moreover, EERPA-GVB is shown to cure a notorious problem of uncorrelated electron-pair models, namely, spatial symmetry breaking in aromatic molecules, using the example of benzene. We have also successfully applied EERPA-GVB to a challenging problem of a phase transition of the boron chain system, where the correlation changes its character along the reaction path. The accuracy and versatility of EERPA-GVB are accompanied by its attractively low computational cost. By truncation of the extended RPA equations and consideration of only at most two-fragment correlation contributions, the cost of computing the EERPA correlation energy is reduced to scale only quadratically with the number of pairs of electrons.

A fractional diffusion random laser

The goal of this letter is to introduce the concept of a non-resonant fractional random laser. This is achieved by extending the classical Letokhov model of photon diffusion through disordered gain media to fractional differential operators in space and time. Fractional transport equations effectively describe anomalous photon sub-diffusion phenomena in non-uniform random scattering media with memory and long-range spatial correlation effects. In particular, by analytically solving fractional transport equations in the one-dimensional slab geometry we obtain simple closed-form expressions for the critical amplification volumes required to initiate the laser action in both fractional-order (FO) and distributed-order (DO) space-time fractional reaction-diffusion equations. Our findings demonstrate the benefits of anomalous sub-diffusive photon transport in active media with correlated disorder and stimulate the engineering of novel non-resonant random lasers with significantly reduced footprint and amplification volumes beyond the limitations of uniform disorder and Markovian diffusion processes.

Chemisorption of carbon monoxide, carbon dioxide and “methanephobia” on the [0001] titanium surface

We performed simulations with density functional theory and molecular dynamics to investigate the interaction of CO, CO2, and CH4 with the surface of a [0001] titanium slab made of four titanium layers. Our simulations were performed at atmospheric pressure and 300 K. We found that the CO molecule is chemisorbed, with dissociation at the surface with an adsorption energy of −3.11 eV. The CO2 molecule can be physisorbed or chemisorbed. In physisorption, the molecule ends up at a distance from the surface of around 3.8 Å. The adsorption energy is −0.3157 eV. The chemisorption occurs in three possible ways. In the first, the molecule is adsorbed with no dissociation in the process, with an adsorption energy of −3.157 eV. In the second, the molecule is dissociated to CO and O, with an adsorption energy of ‐5.2028 eV. In the third case, the dissociation is complete, and the adsorption energy is −8.595 eV. We also used the nudged-elastic-band (NEB) method to calculate the chemisorption energies for this molecule, and we found very similar results. Finally, it is found that the CH4 molecule is repelled from the surface. These calculations were performed these calculations using standard GGA, and afterwards we utilized Vdw-DF2, which includes long-range correlation effects as Van der Waals interactions. The same results are obtained. It can be said that the [0001] titanium surface presents “methane phobia.”

Understanding and Quantifying London Dispersion Effects in Organometallic Complexes.

Quantum chemical methods are nowadays able to determine properties of larger chemical systems with high accuracy and Kohn-Sham density functional theory (DFT) in particular has proven to be robust and suitable for everyday applications of electronic structure theory. A clear disadvantage of many established standard density functional approximations like B3LYP is their inability to describe long-range electron correlation effects. The inclusion of such effects, also termed London dispersion, into DFT has been extensively researched in recent years, resulting in some efficient and routinely used correction schemes. The well-established D3 method has demonstrated its efficiency and accuracy in numerous applications since 2010. Recently, it was improved by developing the successor (termed D4) which additionally includes atomic partial charge information for the generation of pairwise dispersion coefficients. These coefficients determine the leading-order (two-body) and higher-order (three- or many-body) terms of the D4 dispersion energy which is simply added to a standard DFT energy. With its excellent accuracy-to-cost ratio, the DFT-D4 method is well suited for the determination of structures and chemical properties for molecules of most kinds. While dispersion effects in organic molecules are nowadays well studied, much less is known for organometallic complexes. For such systems, there has been a growing interest in designing dispersion-controlled reactions especially in the field of homogeneous catalysis. Here, efficient electronic structure methods are necessary for screening of promising model complexes and quantifying dispersion effects. In this Account, we describe the quality of calculated structural and thermodynamic properties in gas-phase obtained with DFT-D4 corrected methods, specifically for organometallic complexes. The physical effects leading to London dispersion interactions are briefly discussed in the picture of second-order perturbation theory. Subsequently, basic theoretical aspects of the D4 method are introduced followed by selected case studies. Several chemical examples are presented starting with the analysis of transition metal thermochemistry and noncovalent interactions for small, heavy element containing main group compounds. Computed reaction energies can only match highly accurate reference values when all energy contributions are included in the DFT treatment, thus highlighting the major role of dispersion interactions for the accurate description of thermochemistry in gas-phase. Furthermore, the correlation between structural and catalytic properties is emphasized where the accessibility of high quality structures is essential for reaction planning and catalyst design. We present calculations for aggregates of organometallic systems with intrinsically large repulsive electrostatic interactions which can be stabilized by London dispersion effects. The newly introduced inclusion of atomic charge information in the DFT-D4 model robustly leads to quantitatively improved dispersion energies in particular for metallic systems. By construction it yields results which are easily understandable due to a clear separation into hybridization and charge (oxidation) state and two- and many-body effects, respectively. Due to its high computational efficiency, the D4 dispersion model is even applicable to low-cost classical and semiempirical theoretical methods.

Accurate Predictions of Electron Binding Energies of Dipole-Bound Anions via Quantum Monte Carlo Methods.

Neutral molecules with sufficiently large dipole moments can bind electrons in diffuse nonvalence orbitals with most of their charge density far from the nuclei, forming so-called dipole-bound anions. Because long-range correlation effects play an important role in the binding of an excess electron and overall binding energies are often only on the order of 10s-100s of wave numbers, predictively modeling dipole-bound anions remains a challenge. Here, we demonstrate that quantum Monte Carlo methods can accurately characterize molecular dipole-bound anions with near-threshold dipole moments. We also show that correlated sampling Auxiliary Field Quantum Monte Carlo is particularly well-suited for resolving the fine energy differences between the neutral and anionic species. These results shed light on the fundamental limitations of quantum Monte Carlo methods and pave the way toward using them for the study of weakly bound species that are too large to model using traditional electron structure methods.

Theoretical approaches for treating non-valence correlation-bound anions

In this work, we use a model (H2O)4 cluster, the bent CO2 molecule, and tetracyanoethylene as systems to explore the applicability of various electronic structure methods for characterizing non-valence correlation-bound anion states. The methods examined include the algebraic diagrammatic construction, various equation-of-motion coupled cluster methods, orbital-optimized MP2, and Brueckner coupled cluster doubles with perturbative triples. We demonstrate that the key to treating this challenging class of anions is the use of methods that include adequate orbital relaxation in response to long-range dispersion-like correlation effects.

Photoactivated processes in optical fibers: generation and conversion mechanisms of twofold coordinated Si and Ge atoms

In this work we present an extensive investigation of nanoscale physical phenomena related to oxygen-deficient centers (ODCs) in silica and Ge-doped silica by means of first-principles calculations, including nudged-elastic band, electron paramagnetic resonance parameters calculations, and many-body perturbation theory (GW and Bethe–Salpeter equation) techniques. We show that by neutralizing positively charged oxygen monovacancies we can obtain model structures of twofold Si and Ge defects of which the calculated absorption spectra and singlet-to-triplet transitions are in excellent agreement with the experimental optical absorption and photo-luminescence data. In particular we provide an exhaustive analysis of the main exciton peaks related to the presence of twofold defects including long-range correlation effects. By calculating the reaction pathways and energy barriers necessary for the interconversion, we advance a double precursory origin of the and Ge(2) centers as due to the ionization of neutral oxygen monovacancies (Si–Si and Ge–Si dimers) and as due to the ionization of twofold Si and Ge defects. Furthermore two distinct structural conversion mechanisms are found to occur between the neutral oxygen monovacancy and the twofold Si (and Ge) atom configurations. Such conversion mechanisms allow to explain the radiation induced generation of the ODC(II) centers, their photobleaching, and also their generation during the drawing of optical fibers.

Changes in the reflectivity of a lithium niobate crystal decorated with a graphene layer

Density functional theory and molecular dynamics were used to study the interaction of a graphene layer with the surface of lithium niobate. The simulations were performed at atmospheric pressure and 300K. We found that the graphene layer is physisorbed with an adsorption energy of -0.8205 eV/C-atom. Subsequently, the optical absorption of the graphene-(lithium niobate) system was calculated and compared with that of graphene solo and lithium niobate alone, respectively. The calculations were performed using the Quantum Espresso code with the GGA approximation and Vdw-DF2 (which includes long-range correlation effects as Van der Waals interactions).

A Study of Memory Effects in a Chess Database.

A series of recent works studying a database of chronologically sorted chess games–containing 1.4 million games played by humans between 1998 and 2007– have shown that the popularity distribution of chess game-lines follows a Zipf’s law, and that time series inferred from the sequences of those game-lines exhibit long-range memory effects. The presence of Zipf’s law together with long-range memory effects was observed in several systems, however, the simultaneous emergence of these two phenomena were always studied separately up to now. In this work, by making use of a variant of the Yule-Simon preferential growth model, introduced by Cattuto et al., we provide an explanation for the simultaneous emergence of Zipf’s law and long-range correlations memory effects in a chess database. We find that Cattuto’s Model (CM) is able to reproduce both, Zipf’s law and the long-range correlations, including size-dependent scaling of the Hurst exponent for the corresponding time series. CM allows an explanation for the simultaneous emergence of these two phenomena via a preferential growth dynamics, including a memory kernel, in the popularity distribution of chess game-lines. This mechanism results in an aging process in the chess game-line choice as the database grows. Moreover, we find burstiness in the activity of subsets of the most active players, although the aggregated activity of the pool of players displays inter-event times without burstiness. We show that CM is not able to produce time series with bursty behavior providing evidence that burstiness is not required for the explanation of the long-range correlation effects in the chess database. Our results provide further evidence favoring the hypothesis that long-range correlations effects are a consequence of the aging of game-lines and not burstiness, and shed light on the mechanism that operates in the simultaneous emergence of Zipf’s law and long-range correlations in a community of chess players.

Differential operator multiplication method for fractional differential equations

Fractional derivatives play a very important role in modeling physical phenomena involving long-range correlation effects. However, they raise challenges of computational cost and memory storage requirements when solved using current well developed numerical methods. In this paper, the differential operator multiplication method is proposed to address the issues by considering a reaction---advection---diffusion equation with a fractional derivative in time. The linear fractional differential equation is transformed into an integer order differential equation by the proposed method, which can fundamentally fix the aforementioned issues for select fractional differential equations. In such a transform, special attention should be paid to the initial conditions for the resulting differential equation of higher integer order. Through numerical experiments, we verify the proposed method for both fractional ordinary differential equations and partial differential equations.

Electronic properties of molecules and surfaces with a self-consistent interatomic van der Waals density functional.

How strong is the effect of van der Waals (vdW) interactions on the electronic properties of molecules and extended systems? To answer this question, we derived a fully self-consistent implementation of the density-dependent interatomic vdW functional of Tkatchenko and Scheffler [Phys. Rev. Lett. 102, 073005 (2009)]. Not surprisingly, vdW self-consistency leads to tiny modifications of the structure, stability, and electronic properties of molecular dimers and crystals. However, unexpectedly large effects were found in the binding energies, distances, and electrostatic moments of highly polarizable alkali-metal dimers. Most importantly, vdW interactions induced complex and sizable electronic charge redistribution in the vicinity of metallic surfaces and at organic-metal interfaces. As a result, a substantial influence on the computed work functions was found, revealing a nontrivial connection between electrostatics and long-range electron correlation effects.

Electrostatic Domination of the Effect of Electron Correlation in Intermolecular Interactions.

The electron-electron correlation energy is negative, and attractive dispersion interactions are entirely a correlation effect; therefore, the contribution of correlation to intermolecular binding is commonly assumed to be negative, or binding in nature. However, there are many cases where the long-range correlation binding energy is positive, with certain geometries of the water dimer as a prominent example. Geometries with dipoles misaligned can also have an electrostatically dominated, though negative, long-range correlation binding. In either case, the interaction decays as R(-3). This has its origin in the systematic overestimation of dipole moments by Hartree-Fock theory, leading to a reduction in the calculated electrostatic attraction upon inclusion of correlation. Thus, energy decomposition analyses that include correlation but do not correct mean field electrostatic terms are suboptimal. Attenuated second-order Møller-Plesset theory, which smoothly truncates long-range electron correlation effects to zero, can, paradoxically, have the correct long-range behavior for many intermolecular interactions.

Long-range correlation effects in directional living polymers.

The statistics of (equilibrium) living polymers including both linear chains and rings are considered theoretically. Particular attention is addressed to directional polymers characterized by an arrow along the backbone defined by its chemical structure. Thermodynamic and correlation properties of living polymers are studied both in the mean-field and in the critical scaling regimes. It is shown that living polymers with no rings, classical living polymers with rings, and directional living polymers with rings form three distinct classes characterized by different critical exponents and qualitatively different long-range correlation functions.

Geometrical Correction for the Inter- and Intramolecular Basis Set Superposition Error in Periodic Density Functional Theory Calculations

We extend the previously developed geometrical correction for the inter- and intramolecular basis set superposition error (gCP) to periodic density functional theory (DFT) calculations. We report gCP results compared to those from the standard Boys-Bernardi counterpoise correction scheme and large basis set calculations. The applicability of the method to molecular crystals as the main target is tested for the benchmark set X23. It consists of 23 noncovalently bound crystals as introduced by Johnson et al. (J. Chem. Phys. 2012, 137, 054103) and refined by Tkatchenko et al. (J. Chem. Phys. 2013, 139, 024705). In order to accurately describe long-range electron correlation effects, we use the standard atom-pairwise dispersion correction scheme DFT-D3. We show that a combination of DFT energies with small atom-centered basis sets, the D3 dispersion correction, and the gCP correction can accurately describe van der Waals and hydrogen-bonded crystals. Mean absolute deviations of the X23 sublimation energies can be reduced by more than 70% and 80% for the standard functionals PBE and B3LYP, respectively, to small residual mean absolute deviations of about 2 kcal/mol (corresponding to 13% of the average sublimation energy). As a further test, we compute the interlayer interaction of graphite for varying distances and obtain a good equilibrium distance and interaction energy of 6.75 Å and -43.0 meV/atom at the PBE-D3-gCP/SVP level. We fit the gCP scheme for a recently developed pob-TZVP solid-state basis set and obtain reasonable results for the X23 benchmark set and the potential energy curve for water adsorption on a nickel (110) surface.

Many‐Body Dispersion Interactions in Molecular Crystal Polymorphism

Polymorphs of molecular crystals are often very close in energy, yet they may possess very different physical and chemical properties. The understanding of polymorphism is therefore of great importance for a variety of applications, ranging from drug design to nonlinear optics and hydrogen storage. While the crystal structure prediction blind tests conducted by the Cambridge Crystallographic Data Centre have shown steady progress toward reliable structure prediction for molecular crystals, several challenges remain, including molecular salts, hydrates, and flexible molecules with several stable conformers. The ability to identify and rank all of the relevant polymorphs of a given molecular crystal hinges on an accurate description of their relative energetic stability. Hence, a first-principles quantum mechanical method that can attain the required accuracy of around 0.1–0.2 kcalmol 1 would clearly be an indispensable tool for polymorph prediction. In this work, we show that accounting for the nonadditive many-body dispersion (MBD) energy beyond the standard pairwise approximation is crucial for the correct qualitative and quantitative description of polymorphism in molecular crystals. We demonstrate this through three fundamental and stringent benchmark examples: glycine, oxalic acid, and tetrolic acid. These systems represent a broad class of molecular crystals, comprising hydrogenbonded (H-bonded) networks of amino acids and carboxylic acids. Among the first-principles methods, density functional theory (DFT) is the most widely used approach in the study of polymorphism in molecular crystals. However, most common exchange-correlation functionals (including hybrid functionals) are based on semi-local electron correlation, and thereby fail to capture the contribution of dispersion interactions to the stability of molecular crystals. These ubiquitous noncovalent interactions are quantum mechanical in nature and correspond to the multipole moments induced in response to instantaneous fluctuations in the electron density. To incorporate these long-range electron correlation effects within DFT, significant progress has been made by using the standard C6/R 6 pairwise additive expression for the dispersion energy. Indeed, DFT with pairwise dispersion terms can yield accurate results when the energy differences between molecular crystal polymorphs are sufficiently large. Notably, Neumann et al. have achieved the highest success rate in the last two blind tests using such methods. However, these pairwise dispersion approaches, even when used in conjunction with state-of-the-art functionals, are still unable to reach the level of accuracy required to describe polymorphism in many relevant molecular crystals, including glycine and oxalic acid. Recently, a novel and efficient method for describing the many-body dispersion (MBD) energy has been developed, building upon the Tkatchenko–Scheffler (TS) pairwise method. Within the TS approach, the effective dispersion coefficients (C6) are calculated from the DFTelectron density, hence the effect of the local environment of an atom in a molecule is accurately accounted for by construction. The MBD method presents a two-fold improvement over the TS approach by including: 1) the long-range electrodynamic screening through the self-consistent solution of the dipole– dipole electric-field coupling equations for the effective polarizability, and 2) the non-pairwise-additive many-body dispersion energy to infinite order through diagonalization of the Hamiltonian corresponding to a system of coupled fluctuating dipoles. The inclusion of the MBD energy in DFT leads to a significant improvement in the binding energies between organic molecules, and for the cohesion of the benzene and oligoacene molecular crystals. The MBD energy, like the TS energy, can be added to any DFT functional, requiring only the adjustment of a single rangeseparation parameter per functional. [*] N. Marom, J. R. Chelikowsky Center for Computational Materials Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, TX 78712 (USA) E-mail: noa@ices.utexas.edu