Basis Set Size Research Articles

Precise all-electron dynamical response functions: Application to COHSEX and the RPA correlation energy

We present a methodology to calculate frequency and momentum dependent all-electron response functions determined within Kohn-Sham density functional theory. It overcomes the main obstacle in calculating response functions in practice, which is the slow convergence with respect to the number of unoccupied states and the basis-set size. In this approach, the usual sum-over-states expression of perturbation theory is complemented by the response of the orbital basis functions, explicitly constructed by radial integrations of frequency-dependent Sternheimer equations. To an essential extent an infinite number of unoccupied states are included in this way. Furthermore, the response of the core electrons is treated virtually exactly, which is out of reach otherwise. The method is an extension of the recently introduced incomplete-basis-set correction (IBC) [Betzinger et al., Phys. Rev. B85, 245124 (2012); 88, 075130 (2013)] to the frequency and momentum domain. We have implemented the generalized IBC within the all-electron full-potential linearized augmented-plane-wave method and demonstrate for rocksalt BaO the improved convergence of the dynamical Kohn-Sham polarizability. We apply this technique to compute (a) quasiparticle energies employing the COHSEX approximation for the self-energy of many-body perturbation theory and (b) all-electron RPA correlation energies. It is shown that the favorable convergence of the polarizability is passed over to the COHSEX and RPA calculation.

Many interacting fermions in a one-dimensional harmonic trap: a quantum-chemical treatment

We employ ab initio methods of quantum chemistry to investigate spin-1/2 fermions interacting via a two-body contact potential in a one-dimensional harmonic trap. The convergence of the total energy with the size of the one-particle basis set is analytically investigated for the two-body problem and the same form of the convergence formula is numerically confirmed to be valid for the many-body case. Benchmark calculations for two to six fermions with the full configuration interaction method equivalent to the exact diagonalization approach, and the coupled cluster (CC) method including single, double, triple, and quadruple excitations are presented. The convergence of the correlation energy with the level of excitations included in the CC model is analyzed. The range of the interaction strength for which single-reference CC methods work is examined. Next, the CC method restricted to single, double, and noniterative triple excitations, CCSD(T), is employed to study a two-component Fermi gas composed of 6–80 atoms in a one-dimensional harmonic trap. The density profiles of trapped atomic clouds are also reported. Finally, a comparison with experimental results for few-fermion systems is presented. Upcoming possible applications and extensions of the presented approach are discussed.

Mechanical Properties from Periodic Plane Wave Quantum Mechanical Codes: The Challenge of the Flexible Nanoporous MIL-47(V) Framework

Modeling the flexibility of metal–organic frameworks (MOFs) requires the computation of mechanical properties from first principles, e.g., for screening of materials in a database, for gaining insight into structural transformations, and for force field development. However, this paper shows that computations with periodic density functional theory are challenged by the flexibility of these materials: guidelines from experience with standard solid-state calculations cannot be simply transferred to flexible porous frameworks. Our test case, the MIL-47(V) material, has a large-pore and a narrow-pore shape. The effect of Pulay stress (cf. Pulay forces) leads to drastic errors for a simple structure optimization of the flexible MIL-47(V) material. Pulay stress is an artificial stress that tends to lower the volume and is caused by the finite size of the plane wave basis set. We have investigated the importance of this Pulay stress, of symmetry breaking, and of k-point sampling on (a) the structure optimization and (b) mechanical properties such as elastic constants and bulk modulus, of both the large-pore and narrow-pore structure of MIL-47(V). We found that, in the structure optimization, Pulay effects should be avoided by using a fitting procedure, in which an equation of state E(V) (EOS) is fit to a series of energy versus volume points. Manual symmetry breaking could successfully lower the energy of MIL-47(V) by distorting the vanadium–oxide distances in the vanadyl chains and by rotating the benzene linkers. For the mechanical properties, the curvature of the EOS curve was compared with the Reuss bulk modulus, derived from the elastic tensor in the harmonic approximation. Errors induced by anharmonicity, the eggbox effect, and Pulay effects propagate into the Reuss modulus. The strong coupling of the unit cell axes when the unit cell deforms expresses itself in numerical instability of the Reuss modulus. For a flexible material, it is therefore advisible to resort to the EOS fit procedure.

Optical properties of alkali halide crystals from all-electron hybrid TD-DFT calculations.

We present a study of the electronic and optical properties of a series of alkali halide crystals AX, with A = Li, Na, K, Rb and X = F, Cl, Br based on a recent implementation of hybrid-exchange time-dependent density functional theory (TD-DFT) (TD-B3LYP) in the all-electron Gaussian basis set code CRYSTAL. We examine, in particular, the impact of basis set size and quality on the prediction of the optical gap and exciton binding energy. The formation of bound excitons by photoexcitation is observed in all the studied systems and this is shown to be correlated to specific features of the Hartree-Fock exchange component of the TD-DFT response kernel. All computed optical gaps and exciton binding energies are however markedly below estimated experimental and, where available, 2-particle Green's function (GW-Bethe-Salpeter equation, GW-BSE) values. We attribute this reduced exciton binding to the incorrect asymptotics of the B3LYP exchange correlation ground state functional and of the TD-B3LYP response kernel, which lead to a large underestimation of the Coulomb interaction between the excited electron and hole wavefunctions. Considering LiF as an example, we correlate the asymptotic behaviour of the TD-B3LYP kernel to the fraction of Fock exchange admixed in the ground state functional cHF and show that there exists one value of cHF (∼0.32) that reproduces at least semi-quantitatively the optical gap of this material.

Two new methods for computing vibrational energy levels

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential. The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix–vector products.

Converging nuclear magnetic shielding calculations with respect to basis and system size in protein systems

Ab initio chemical shielding calculations greatly facilitate the interpretation of nuclear magnetic resonance (NMR) chemical shifts in biological systems, but the large sizes of these systems requires approximations in the chemical models used to represent them. Achieving good convergence in the predicted chemical shieldings is necessary before one can unravel how other complex structural and dynamical factors affect the NMR measurements. Here, we investigate how to balance trade-offs between using a better basis set or a larger cluster model for predicting the chemical shieldings of the substrates in two representative examples of protein-substrate systems involving different domains in tryptophan synthase: the N-(4'-trifluoromethoxybenzoyl)-2-aminoethyl phosphate (F9) ligand which binds in the α active site, and the 2-aminophenol quinonoid intermediate formed in the β active site. We first demonstrate that a chemically intuitive three-layer, locally dense basis model that uses a large basis on the substrate, a medium triple-zeta basis to describe its hydrogen-bonding partners and/or surrounding van der Waals cavity, and a crude basis set for more distant atoms provides chemical shieldings in good agreement with much more expensive large basis calculations. Second, long-range quantum mechanical interactions are important, and one can accurately estimate them as a small-basis correction to larger-basis calculations on a smaller cluster. The combination of these approaches enables one to perform density functional theory NMR chemical shift calculations in protein systems that are well-converged with respect to both basis set and cluster size.

Choosing an atomic basis set for TD-DFT, SOPPA, ADC(2), CIS(D), CC2 and EOM-CCSD calculations of low-lying excited states of organic dyes

Aiming to pinpoint an atomic basis set providing accurate transition energies at a minimal computational cost, we investigate the evolution with basis set size of the energy of low-lying excited states in nine representative conjugated dyes with a wide panel of theoretical approaches, namely TD-DFT, SOPPA, ADC(2), CIS(D), CC2, EOM-CCSD, as well as several scaled opposite spin variants, namely SOS-CIS(D), SOS-CIS(D0) and SOS-ADC(2). At the exception of TD-DFT that displays the lowest basis set dependence, it turns out that the changes obtained when increasing the size of the basis set are rather independent of the selected wavefunction model, but strongly change according to the nature of the excited state considered. Reasonable compromises between accuracy and computational burden can be attained with 6-311+G(2d,p) that allows much faster calculations than the typical reference basis set, namely aug-cc-pVTZ, for an average loss of accuracy limited to ca. 0.02 eV.

Estimating the CCSD basis-set limit energy from small basis sets: basis-set extrapolations vs additivity schemes

Coupled cluster calculations with all single and double excitations (CCSD) converge exceedingly slowly with the size of the one-particle basis set. We assess the performance of a number of approaches for obtaining CCSD correlation energies close to the complete basis-set limit in conjunction with relatively small DZ and TZ basis sets. These include global and system-dependent extrapolations based on the A + B/Lα two-point extrapolation formula, and the well-known additivity approach that uses an MP2-based basis-set-correction term. We show that the basis set convergence rate can change dramatically between different systems(e.g.it is slower for molecules with polar bonds and/or second-row elements). The system-dependent basis-set extrapolation scheme, in which unique basis-set extrapolation exponents for each system are obtained from lower-cost MP2 calculations, significantly accelerates the basis-set convergence relative to the global extrapolations. Nevertheless, we find that the simple MP2-based basis-set additivity scheme outperforms the extrapolation approaches. For example, the following root-mean-squared deviations are obtained for the 140 basis-set limit CCSD atomization energies in the W4-11 database: 9.1 (global extrapolation), 3.7 (system-dependent extrapolation), and 2.4 (additivity scheme) kJ mol–1. The CCSD energy in these approximations is obtained from basis sets of up to TZ quality and the latter two approaches require additional MP2 calculations with basis sets of up to QZ quality. We also assess the performance of the basis-set extrapolations and additivity schemes for a set of 20 basis-set limit CCSD atomization energies of larger molecules including amino acids, DNA/RNA bases, aromatic compounds, and platonic hydrocarbon cages. We obtain the following RMSDs for the above methods: 10.2 (global extrapolation), 5.7 (system-dependent extrapolation), and 2.9 (additivity scheme) kJ mol–1.

Quantum dynamics of tunneling dominated reactions at low temperatures

We report a quantum dynamics study of the Li + HF → LiF + H reaction at low temperatures of interest to cooling and trapping experiments. Contributions from non-zero partial waves are analyzed and results show narrow resonances in the energy dependence of the cross section that survive partial wave summation. The computations are performed using the ABC code and a simple modification of the ABC code that enables separate energy cutoffs for the reactant and product rovibrational energy levels is found to dramatically reduce the basis set size and computational expense. Results obtained using two ab initio electronic potential energy surfaces for the LiHF system show strong sensitivity to the choice of the potential. In particular, small differences in the barrier heights of the two potential surfaces are found to dramatically influence the reaction cross sections at low energies. Comparison with recent measurements of the reaction cross section (Bobbenkamp et al 2011 J. Chem. Phys. 135 204306) shows similar energy dependence in the threshold regime and an overall good agreement with experimental data compared to previous theoretical results. Also, usefulness of a recently introduced method for ultracold reactions that employ the quantum close-coupling method at short-range and the multichannel quantum defect theory at long-range, is demonstrated in accurately evaluating product state-resolved cross sections for D + H2 and H + D2 reactions.

Linear scaling density matrix real time TDDFT: Propagator unitarity and matrix truncation.

Real time, density matrix based, time dependent density functional theory (TDDFT) proceeds through the propagation of the density matrix, as opposed to the Kohn-Sham orbitals. It is possible to reduce the computational workload by imposing spatial cutoff radii on sparse matrices, and the propagation of the density matrix in this manner provides direct access to the optical response of very large systems, which would be otherwise impractical to obtain using the standard formulations of TDDFT. Following a brief summary of our implementation, along with several benchmark tests illustrating the validity of the method, we present an exploration of the factors affecting the accuracy of the approach. In particular, we investigate the effect of basis set size and matrix truncation, the key approximation used in achieving linear scaling, on the propagator unitarity and optical spectra. Finally, we illustrate that, with an appropriate density matrix truncation range applied, the computational load scales linearly with the system size and discuss the limitations of the approach.

Efficient Self-Consistent Implementation of Local Hybrid Functionals.

Local hybrid density functionals, with position-dependent exact-exchange admixture, are an important extension to the popular global hybrid functionals, promising improved accuracy for many properties. An efficient implementation is crucial to make local hybrids available for widespread application. The resolution-of-the-identity approach used in previous implementations to compute nonstandard two-electron integrals has been found to require large uncontracted basis sets, rendering the cost of local hybrid calculations impractical for large-scale systems. On the basis of recently promoted seminumerical implementations of exact exchange in global hybrid functionals, we present an efficient, self-consistent implementation of local hybrid functionals within the generalized Kohn-Sham scheme. The final cost of a local hybrid calculation is equal to that of a meta-GGA global hybrid using the seminumerical algorithm. Since seminumerical schemes exhibit superior scaling with respect to system and basis set size over analytical exact exchange, and this advantage is not affected by a position-dependent admixture of exact exchange, local hybrid calculations for large systems are now possible.

Importance and Reliability of Small Basis Set CCSD(T) Corrections to MP2 Binding and Relative Energies of Water Clusters

MP2 describes hydrogen-bonded systems well, yet a higher-order electron correlation correction in the form of a CCSD(T) calculation is usually necessary to achieve benchmark quality energies. We evaluated the importance and reliability of small basis set CCSD(T) corrections to MP2 (δ(MP2)(CCSD(T))) both on the binding (ΔE) and relative (ΔΔE) energies for a large number of systems including four water dimer stationary points and 57 other clusters up to undecamers, (H2O)11. By comparing the MP2 energies with CCSD(T) and the explicitly correlated MP2-F12 energies with variants of CCSD(T)-F12 using different basis sets, we were able to establish that the correction to the binding energy (ΔE) is sensitive to basis set size, especially for small double-ζ basis sets. On the other hand, the basis set sensitivity of the correction to the relative energy (ΔΔE) within each cluster size is very small. While the δ(MP2)(CCSD(T)) correction to the binding energy might vary in magnitude with basis set size, its effect on relative energy (and hence the stability of isomers) is remarkably consistent. Therefore, we recommend the inclusion of this correction to obtain the relative stability of closely spaced isomers using a double-ζ basis set with polarization and diffuse functions such as aug-cc-pVDZ.

Ab initio O(N) elongation-counterpoise method for BSSE-corrected interaction energy analyses in biosystems

An Elongation-counterpoise (ELG-CP) method was developed for performing accurate and efficient interaction energy analysis and correcting the basis set superposition error (BSSE) in biosystems. The method was achieved by combining our developed ab initio O(N) elongation method with the conventional counterpoise method proposed for solving the BSSE problem. As a test, the ELG-CP method was applied to the analysis of the DNAs' inter-strands interaction energies with respect to the alkylation-induced base pair mismatch phenomenon that causes a transition from G⋯C to A⋯T. It was found that the ELG-CP method showed high efficiency (nearly linear-scaling) and high accuracy with a negligibly small energy error in the total energy calculations (in the order of 10(-7)-10(-8) hartree/atom) as compared with the conventional method during the counterpoise treatment. Furthermore, the magnitude of the BSSE was found to be ca. -290 kcal/mol for the calculation of a DNA model with 21 base pairs. This emphasizes the importance of BSSE correction when a limited size basis set is used to study the DNA models and compare small energy differences between them. In this work, we quantitatively estimated the inter-strands interaction energy for each possible step in the transition process from G⋯C to A⋯T by the ELG-CP method. It was found that the base pair replacement in the process only affects the interaction energy for a limited area around the mismatch position with a few adjacent base pairs. From the interaction energy point of view, our results showed that a base pair sliding mechanism possibly occurs after the alkylation of guanine to gain the maximum possible number of hydrogen bonds between the bases. In addition, the steps leading to the A⋯T replacement accompanied with replications were found to be unfavorable processes corresponding to ca. 10 kcal/mol loss in stabilization energy. The present study indicated that the ELG-CP method is promising for performing effective interaction energy analyses in biosystems.

First-Principles Calculations of the Energy and Width of the (2)A(u) Shape Resonance in p-Benzoquinone: A Gateway State for Electron Transfer.

Quinones are versatile biological electron acceptors and mobile electron carriers in redox processes. We present the first ab initio calculations of the width of the (2)A(u) shape resonance in the para-benzoquinone anion, the simplest member of the quinone family. This resonance state located at 2.5 eV above the ground state of the anion is believed to be a gateway state for electron attachment in redox processes involving quinones. We employ the equation-of-motion coupled-cluster method for electron affinity augmented by a complex-absorbing potential (CAP-EOM-EA-CCSD) to calculate the resonance position and width. The calculated width, 0.013 eV, is in excellent agreement with the width of the resonant peak in the photodetachment spectrum, thus supporting the assignment of the band to resonance excitation to the autodetaching (2)A(u) state. The methodological aspects of CAP-EOM-EA-CCSD calculations of resonances positions and widths in medium-sized molecules, such as basis set and CAP box size effects, are also discussed.

On the validity of the basis set superposition error and complete basis set limit extrapolations for the binding energy of the formic acid dimer.

We report the variation of the binding energy of the Formic Acid Dimer with the size of the basis set at the Coupled Cluster with iterative Singles, Doubles and perturbatively connected Triple replacements [CCSD(T)] level of theory, estimate the Complete Basis Set (CBS) limit, and examine the validity of the Basis Set Superposition Error (BSSE)-correction for this quantity that was previously challenged by Kalescky, Kraka, and Cremer (KKC) [J. Chem. Phys. 140, 084315 (2014)]. Our results indicate that the BSSE correction, including terms that account for the substantial geometry change of the monomers due to the formation of two strong hydrogen bonds in the dimer, is indeed valid for obtaining accurate estimates for the binding energy of this system as it exhibits the expected decrease with increasing basis set size. We attribute the discrepancy between our current results and those of KKC to their use of a valence basis set in conjunction with the correlation of all electrons (i.e., including the 1s of C and O). We further show that the use of a core-valence set in conjunction with all electron correlation converges faster to the CBS limit as the BSSE correction is less than half than the valence electron/valence basis set case. The uncorrected and BSSE-corrected binding energies were found to produce the same (within 0.1 kcal/mol) CBS limits. We obtain CCSD(T)/CBS best estimates for De = - 16.1 ± 0.1 kcal/mol and for D0 = - 14.3 ± 0.1 kcal/mol, the later in excellent agreement with the experimental value of -14.22 ± 0.12 kcal/mol.

Coupled-coherent-states approach for high-order harmonic generation

In this paper, we report a version of the coupled-coherent-states method which is able to accurately compute the high-order harmonic generation (HHG) spectrum of an electron in a laser field in one dimension by the use of trajectory-guided grids of Gaussian wave packets. It is shown that by periodic reprojection of the wave function and dynamically altering the basis set size, the method can account for a wave function which spreads out to cover a large area in phase space while still keeping computational expense low and ensuring the preservation of coherence of the wave function. The HHG spectra obtained show good agreement with those from a time-dependent Schr\odinger equation solver. We show also that the part of the wave function which is responsible for HHG moves along a periodic orbit which is far from that of classical motion. Although this paper is a proof of principle and therefore focused on a simple one-dimensional system, future generalizations for the multielectron case are discussed.

Chemical bonding in carborane/aromatic co-polymers: a first-principles analysis of experimental photoemission spectra

First-principles density functional theory calculations have been used to study the relative stability and analyse the chemical bonding of novel cross-linked carborane polymers. Atomic charges with several population analysis methods based on fully relaxed structures were calculated to interpret the chemical binding energy shifts of XPS spectra of these boron carbide polymers. The results indicate that a base structure with one aromatic linking unit with carborane is energetically favoured. The linear relationship between experimental core-level photoemission binding energies and computational partial atomic charges from four population analysis methods (Mulliken, Hirshfeld, atoms-in-molecules (AIM) and natural bond order (NBO)) were analysed and the results indicate that cross-linking occurs at icosahedral B sites non-adjacent to icosahedral carbon sites, in agreement with recently reported experimental results. The role of basis set size in determining partial atomic charges was found to vary with population analysis method. Best linear correlations were identified with the more robust population analysis methods (Hirshfeld, AIM and NBO) with the AIM methods noted as being particularly sensitive to basis set size.

Feature of Exact Exchange Kohn–Sham Orbitals with Krieger–Li–Iafrate Approximation#

We report feature of Kohn–Sham (KS) molecular orbitals computed with the Krieger–Li–Iafrate (KLI) approximation for exact exchange through the comparison to the results of Hartree–Fock (HF) and other KS methods such as local density approximation (LDA) and generalized gradient approximation (GGA). KLI occupied orbitals have similar energies and shapes with those of HF. KLI virtual orbitals are likely to form bound states with negative eigenvalues due to the correct −1/r asymptotic behavior of KLI potentials, whereas HF virtual orbitals are mostly unbound. As a result, HF orbitals tend to be diffuse because of their plane‐wave‐like nature, and their energies are highly sensitive to the size of basis set. The energies of LDA/GGA orbitals appear to be upshifted by a constant factor from the KLI results, but they also produce unbound virtual orbitals like HF. The energy gaps between KLI occupied and virtual orbitals are very close to the corresponding experimental excitation energies compared to the other methods. We also show that Brillouin's theorem can be applied to a Slater determinant made of KLI orbitals as a corollary of the KLI approximation.

Robust, basis-set independent method for the evaluation of charge-transfer energy in noncovalent complexes.

Separation of the energetic contribution of charge transfer to interaction energy in noncovalent complexes would provide important insight into the mechanisms of the interaction. However, the calculation of charge-transfer energy is not an easy task. It is not a physically well-defined term, and the results might depend on how it is described in practice. Commonly, the charge transfer is defined in terms of molecular orbitals; in this framework, however, the charge transfer vanishes as the basis set size increases toward the complete basis set limit. This can be avoided by defining the charge transfer in terms of the spatial extent of the electron densities of the interacting molecules, but the schemes used so far do not reflect the actual electronic structure of each particular system and thus are not reliable. We propose a spatial partitioning of the system, which is based on a charge transfer-free reference state, namely superimposition of electron densities of the noninteracting fragments. We show that this method, employing constrained DFT for the calculation of the charge-transfer energy, yields reliable results and is robust with respect to the strength of the charge transfer, the basis set size, and the DFT functional used. Because it is based on DFT, the method is applicable to rather large systems.

Theoretical insight into hydrogen adsorption onto graphene: a first-principles B3LYP-D3 study.

This work investigates hydrogen adsorption onto various graphene flakes such as coronene and coronene-like as suitable models of graphene within the framework of the DFT-B3LYP method. The non-local van der Waals (vdW) density functional (B3LYP-D3) method is used for both structural geometry optimization and total energy estimations. Calculations were carried out for a hydrogen molecule above a coronene surface with both conventional and vdW corrected DFT to investigate how these approaches perform in the case of hydrogen adsorption on a graphene surface. Our first-principles results within the B3LYP-D3/def2-TZVPP model show that hydrogen physisorbs on a coronene surface with an adsorption energy of -5.013 (kJ mol(-1)) which is in good agreement with the experimental value. The influence of the basis set and graphene flake size were also evaluated and the results indicate that these slightly affect the adsorption properties. We found also that it is crucial to use non-local dispersion interactions to get accurate results for hydrogen adsorption on a graphene surface. Furthermore, the co-adsorption of H2 molecules onto the graphene surface was investigated. The results obtained at the B3LYP-D3/def2-TZVP level show that H2 molecules can be physisorbed on both sides of the graphene layer with adsorption properties similar to those for a single surface. Finally, we showed that H2 molecules might be bound to the graphene surface via a bilayer adsorption scheme with weak adsorption energy. Charge population and electron density analysis confirm the weak binding nature of the system under consideration.