Minimal Basis Set Research Articles

Toward Highly Accurate Spin State Energetics in First-Row Transition Metal Complexes: A Combined CASPT2/CC Approach

In previous work on the performance of multiconfigurational second-order perturbation theory (CASPT2) in describing spin state energetics in first-row transition metal systems [ Pierloot et al. J. Chem. Theory Comput. 2017 , 13 , 537 - 553 ], we showed that standard CASPT2 works well for valence correlation but does not describe the metal semicore (3s3p) correlation effects accurately. This failure is partially responsible for the well-known bias toward high-spin states of CASPT2. In this paper, we expand our previous work and show that this bias could be partly removed with a combined CASPT2/CC approach: using high-quality CASPT2 with extensive correlation-consistent basis sets for valence correlation and low-cost CCSD(T) calculations with minimal basis sets for the metal semicore (3s3p) correlation effects. We demonstrate that this approach is efficient by studying the spin state energetics of a series of iron complexes modeling important intermediates in oxidative catalytic processes in chemistry and biochemistry. On the basis of a comparison with bare CCSD(T) results from this and previous work, the average error of the CASPT2/CC approach is estimated at around 2 kcal mol-1 in favor of high spin states.

Modified B function basis sets with generalized hyperbolic cosine functions

Abstract Efficient exponential type basis sets constructed from new hyperbolic cosine type B functions have been used in self consistent field calculations for the ground states of the atoms from Helium to Argon and their ions. Different hyperbolic cosine type functions are incorporated into the B functions to increase the accuracy of atomic orbitals and correctly describe the electronic density of systems. The presented results for the minimal basis sets are shown that the quality of generalized hyperbolic cosine type B functions is one of the most appropriate basis sets of B functions, especially for increasing atomic number. A comparison with the standard B functions, hyperbolic cosine type B functions and the corresponding numerical Hartree-Fock values are given in tables. Our study shows that the modification of B functions can be an efficient way of increasing the accuracy of the atomic and molecular SCF calculations. These basis sets may also be used in the calculation of atomic properties such as cusp condition and density properties of atomic electrons. Some numerical results and comparisons are given to clarify the basis sets quality from the density point of view. An improved description of atomic orbitals based on the use of modified B functions will also play an important role in ion-atom collisions problems, semi-emprical and density functional methods.

Atom-Centered Potentials with Dispersion-Corrected Minimal-Basis-Set Hartree-Fock: An Efficient and Accurate Computational Approach for Large Molecular Systems.

We present a computational methodology based on atom-centered potentials (ACPs) for the efficient and accurate structural modeling of large molecular systems. ACPs are atom-centered one-electron potentials that have the same functional form as effective-core potentials. In recent works, we showed that ACPs can be used to produce a correction to the ground-state wave function and electronic energy to alleviate shortcomings in the underlying model chemistry. In this work, we present ACPs for H, C, N, and O atoms that are specifically designed to predict accurate non-covalent binding energies and inter- and intramolecular geometries when combined with dispersion-corrected Hartree-Fock (HF-D3) and a minimal basis-set (scaled MINI or MINIs). For example, the combined HF-D3/MINIs-ACP method demonstrates excellent performance, with mean absolute errors of 0.36 and 0.28 kcal/mol for the S22x5 and S66x8 benchmark sets, respectively, relative to highly correlated complete-basis-set data. The application of ACPs results in a significant decrease in error compared to uncorrected HF-D3/MINIs for all benchmark sets examined. In addition, HF-D3/MINIs-ACP, has a cost only slightly higher than a minimal-basis-set HF calculation and can be used with any electronic structure program for molecular quantum chemistry that uses Gaussian basis sets and effective-core potentials.

CBS extrapolation in electronic structure pushed to the end: a revival of minimal and sub-minimal basis sets.

The complete basis set (CBS) limit is secluded in calculations of electronic structure, and hence CBS extrapolation draws immediate attention. Of the highest importance is the extremely slow-convergent correlation energy as obtained both from Møller-Plesset perturbation theory (MP2) and the gold standard coupled cluster singles, doubles, and perturbative triples method [CCSD(T)], both picked as the focal point. Because today's massive electronic structure calculations demand schemes for CBS extrapolation from the lowest rungs of the hierarchical staircase, the focus is also on how minimal and subminimal basis sets can help in the endeavour. The above has prompted, for the first time, the development of a reliable analytic variant of the unified singlet- and triplet-pair extrapolation (USTE) scheme, which is next utilized to hierarchize any arbitrary basis set. With all available basis set families for molecular orbital calculations systematized as subminimal, minimal and extended from the fraction of correlation energy that they can recover, the full approach is simple and general, yet reliable, and hereby illustrated for basis sets as small as Huzinaga MINI and as extended as the sophisticated ansatzes that are utilized in modern explicitly correlated MP2-F12 and CCSD(T)-F12 calculations. Extensively tested with up to 26 basis sets from various families, the reported results show that CBS extrapolation from a (subminimal, extended) basis-set pair yields correlation energies that outperform by far those obtained from Kohn-Sham density functional theory (KS DFT). When employing two subminimal basis sets, the results may still outperform KS-DFT but at a drastically smaller computational cost. To further gauge its performance, the new method is utilized to calculate the energies of all 45 isomers of C8H8 recently reported at a high ab initio level, with the present results showing excellent agreement with the available data. The present cost-effective approach may therefore be expected to have a broad impact on chemistry and even materials science.

Introducing a new bond reactivity index: Philicities for natural bond orbitals.

In the present work, a new methodology defined for obtaining reactivity indices (philicities) is proposed. This is based on reactivity functions such as the Fukui function or the dual descriptor, and makes it possible to project the information from reactivity functions onto molecular orbitals, instead of onto the atoms of the molecule (atomic reactivity indices). The methodology focuses on the molecules' natural bond orbitals (bond reactivity indices) because these orbitals have the advantage of being localized, allowing the reaction site of an electrophile or nucleophile to be determined within a very precise molecular region. This methodology provides a "philicity" index for every NBO, and a representative set of molecules has been used to test the new definition. A new methodology has also been developed to compare the "finite difference" and the "frontier molecular orbital" approximations. To facilitate their use, the proposed methodology as well as the possibility of calculating the new indices have been implemented in a new version of UCA-FUKUI software. In addition, condensation schemes based on atomic populations of the "atoms in molecules" theory, the Hirshfeld population analysis, the approximation of Mulliken (with a minimal basis set) and electrostatic potential-derived charges have also been implemented, including the calculation of "bond reactivity indices" defined in previous studies. Graphical abstract A new methodology defined for obtaining bond reactivity indices (philicities) is proposed and makes it possible to project the information from reactivity functions onto molecular orbitals. The proposed methodology as well as the possibility of calculating the new indices have been implemented in a new version of UCA-FUKUI software. In addition, this version can use new atomic condensation schemes and new "utilities" have also been included in this second version.

Use of Low-Cost Quantum Chemistry Procedures for Geometry Optimization and Vibrational Frequency Calculations: Determination of Frequency Scale Factors and Application to Reactions of Large Systems.

We have assessed the performance of a variety of low-cost computational quantum chemistry procedures (semiempirical, pure-DFT, and screened-exchange DFT methods) for computing molecular geometries and thermochemical quantities associated with the vibrational frequencies. Frequency scale factors for zero-point vibrational energies and thermal corrections for 298 K enthalpies and 298 K entropies have been determined. In absolute terms, for small to medium-sized molecules, all procedures perform reasonably well. Semiempirical methods have mean absolute deviations (MADs) of ∼15 kJ mol-1 for total enthalpies and free energies. For DFT procedures, hybrid DFT generally performs better than pure DFT. Remarkably, the N12 pure functional shows very good performances (MADs ∼ 3 kJ mol-1) that are comparable to those for hybrid functionals. An examination of the basis set effect indicates N12/3-21G* and N12/6-31G(d) to be cost-effective for geometry optimization and vibrational frequency calculations, but the use of minimal basis sets leads to very large MADs for the calculated thermochemical quantities. Further testing with reaction energies of large systems shows that, by exploiting cancellation of systematic deviations, although the deviations can be very substantial in absolute terms (>100 kJ mol-1), those for relative energies are markedly reduced (∼10 kJ mol-1). This enables the use of even semiempirical procedures to obtain geometries and vibrational frequencies with reasonable accuracy in cases where the use of more expensive procedures is computationally prohibitive.

Noncovalent Interactions by Fixed-Node Diffusion Monte Carlo: Convergence of Nodes and Energy Differences vs Gaussian Basis-Set Size

Convergence of fixed-node (FN) shape and FN diffusion Monte Carlo (FNDMC) interaction energies is studied vs the Gaussian basis set saturation level in HF and CH4 dimers and one-determinant Slater-Jastrow trial wave functions (ΨT). The tested 25 distinct basis sets obtained by stepwise trimming of aug-VDZ and aug-VTZ bases suggest minimum basis set requirements to achieve reasonable results. A single selected trimmed basis set, about 2 times smaller in size than aug-VTZ, is extensively tested on a set of 12 noncovalent complexes including formic acid dimer, benzene-methane, or coronene-H2. The results indicate that equivalent noncovalent FNDMC energy differences are available at costs lower than assumed before. Additional insights from electron density differences and comparison of dimer vs monomer ΨT nodes explain this observation.

Sliced Basis Density Matrix Renormalization Group for Electronic Structure.

We introduce a hybrid approach to applying the density matrix renormalization group to continuous systems, combining a grid approximation along one direction with a finite Gaussian basis set for the remaining two directions. This approach is especially useful for chainlike molecules, where the grid is used in the long direction. For hydrogen chain systems, the computational time scales approximately linearly with the number of atoms, as we show with near-exact minimal basis set calculations with up to 1000 atoms. The linear scaling comes from both the localization of the basis and a compression method for the long-ranged two-electron interaction. For shorter hydrogen chains, we show results with up to triple-ζ bases.

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.

Quantum Dynamics with Short-Time Trajectories and Minimal Adaptive Basis Sets.

Methods for solving the time-dependent Schrödinger equation via basis set expansion of the wave function can generally be categorized as having either static (time-independent) or dynamic (time-dependent) basis functions. We have recently introduced an alternative simulation approach which represents a middle road between these two extremes, employing dynamic (classical-like) trajectories to create a static basis set of Gaussian wavepackets in regions of phase-space relevant to future propagation of the wave function [J. Chem. Theory Comput., 11, 8 (2015)]. Here, we propose and test a modification of our methodology which aims to reduce the size of basis sets generated in our original scheme. In particular, we employ short-time classical trajectories to continuously generate new basis functions for short-time quantum propagation of the wave function; to avoid the continued growth of the basis set describing the time-dependent wave function, we employ Matching Pursuit to periodically minimize the number of basis functions required to accurately describe the wave function. Overall, this approach generates a basis set which is adapted to evolution of the wave function while also being as small as possible. In applications to challenging benchmark problems, namely a 4-dimensional model of photoexcited pyrazine and three different double-well tunnelling problems, we find that our new scheme enables accurate wave function propagation with basis sets which are around an order-of-magnitude smaller than our original trajectory-guided basis set methodology, highlighting the benefits of adaptive strategies for wave function propagation.

Producing DFT/MM enzyme reaction trajectories from SCC-DFTB/MM driving forces to probe the underlying electronics of a glycosyltransferase reaction.

The SCC-DFTB/MIO/CHARMM free energy surface for a glycosyltransferase, TcTS, is benchmarked against a DFT/MM reaction trajectory using the same CHARMM MM force field ported to the NWChem package. The popular B3LYP functional, against which the MIO parameter set was parameterized is used to optimize TS structures and run DFT reaction dynamics. A novel approach was used to generate reaction forces from a SCC-DFTB/MIO/CHARMM reaction surface to drive B3LYP/6-31G/MM and B3LYP/6-31G(d)/MM reaction trajectories. Although TS structures compare favorably, differences stemming primarily from a minimal basis set approximation prevented a successful 6-31G(d) FEARCF reaction dynamics trajectory. None the less, the dynamic evolution of the B3LYP/6-31G/MM-computed electron density provided an opportunity to perform NBO analysis along the reaction trajectory. Here, we illustrate that a successful ab initio reaction trajectory is computationally accessible when the underlying potential energy function of the semi-empirical method used to produce driving forces is sufficiently close to the ab initio potential. © 2017 Wiley Periodicals, Inc.

Relativistic ab Initio Accurate Atomic Minimal Basis Sets: Quantitative LUMOs and Oriented Quasi-Atomic Orbitals for the Elements Li-Xe.

Valence virtual orbitals (VVOs) are a quantitative and basis set independent method for extracting chemically meaningful lowest unoccupied molecular orbitals (LUMOs). The VVOs are formed based on a singular value decomposition (SVD) with respect to precomputed and internally stored ab initio accurate atomic minimal basis sets (AAMBS) for the atoms. The occupied molecular orbitals and VVOs together form a minimal basis set that can be transformed into orthogonal oriented quasi-atomic orbitals (OQUAOs) that provide a quantitative description of the bonding in a molecular environment. In the present work, relativistic AAMBS are developed that span the full valence orbital space. The impact of using full valence AAMBS for the formation of the VVOs and OQUAOs and the resulting bonding analysis is demonstrated with applications to the cuprous chloride, scandium monofluoride, and nickel silicide diatomic molecules.

Density-based errors in mixed-basis mean-field electronic structure, with implications for embedding and QM/MM methods

We consider mean-field electronic structure calculations with subsystems that employ different atomic-orbital basis sets. A major source of error arises in charge-manifestation reactions (including ionization, electron attachment, or deprotonation) due to electronic density artifacts at the subsystem interface. The underlying errors in the electronic density can be largely eliminated with Fock-matrix corrections or by avoiding the use of a minimal basis set in the low-level region. These corrections succeed by balancing the electronegativity of atoms at the subsystem interface, much as link-atoms in QM/MM calculations rely upon balancing the electronegativity of atoms in the truncated QM region.

Comment on the spherical quantum dot with interaction effects

A series of studies of the lowest [Formula: see text] state of a model of a two-electron quantum dot, that consists of a finite spherical well [see, for example, Sivakami et al., Int. J. Mod. Phys. B 24, 5561 (2010)], exhibited a range of interesting applications. However, the quantity of central interest, the expectation value of the interelectronic repulsion (that must be nonnegative) was calculated erroneously, yielding negative values. In the present note, this expectation value is evaluated with respect to the exact first-order perturbation theory wavefunction (the scheme considered in the papers cited above) as well as with respect to a minimal basis set Gaussian variational wavefunction. Reasonable agreement between the two schemes is confirmed, both yielding the expected positive values of the expectation value of the interelectronic repulsion, over the whole range of the relevant parameters.

Description of non-covalent interactions in SCC-DFTB methods.

We have analyzed the description of non-covalent interactions in multiple variants of the self-consistent charges density functional tight binding (SCC-DFTB) method. While the description of London dispersion can be easily improved by empirical correction, hydrogen bonding poses a much more difficult problem. We have implemented an interaction energy decomposition scheme that allowed us to quantify the error at the level of first-order electrostatic and polarization terms. Both are underestimated because of the monopole approximation used in SCC-DFTB, with the latter being affected also by the use of minimal basis set. Among the methods tested, SCC-DFTB with the empirical D3H4 corrections worked best. To make this correction compatible with the latest development in SCC-DFTB, we have reparameterized it for use with third-order SCC-DFTB with the 3OB parameter set. © 2017 Wiley Periodicals, Inc.

Method Dependence of Proline Ring Flexibility in the Poly-l-Proline Type II Polymer.

We studied the sensitivity of the energetic and geometrical features of the proline ring (pyrrolidine) to the quantum mechanical computational approach by adopting the proline monomer, trimer, and polymer, as simplified collagen protein models. Within the Density Functional Theory (DFT) approach, we tested the ability of different functionals (GGA PBE and the hybrid B3LYP), added with a posteriori empirical dispersion corrections (D), to predict the conformational potential energy surface of the five-membered heterocycle pyrrolidine ring for the above models, dictating the collagen main features. We also compared the DFT-D results with those from the recently proposed cost-effective HF-3c method and our variant HF-3c-027, both based on Hartree-Fock Hamiltonian and Gaussian minimal basis set properly corrected for basis set superposition error, structure deficiencies, and dispersion interactions. We found that dispersion interactions are essential to destabilize specific conformers. While the HF-3c and its HF-3c-027 variant are unreliable to predict accurately the energy of the ring conformers, structures are accurate. Indeed, the cost-effective DFT-D//HF-3c-027 approach in which the energetic is from the accurate DFT-D method on HF-3c-027 structures provides energetic in line with that derived by the costly DFT-D//DFT-D approach, paving the way to simulate more realistic collagen models of much larger size. The adoption of either PBE or B3LYP functional for the electronic part of the DFT-D method gives very similar results, recommending the first as the most cost-effective method for simulating large collagen models. The predicted most stable conformation computed for the periodic poly proline (type II) model allows for a higher flexibility, in agreement with experimental studies on collagen protein. The present results open the way to large-scale calculations of the collagen/hydroxyapatite system, crucial for understanding the atomistic details in bones and teeth.

Fundamental relationships between elementary functionals in quantum chemistry. An application for H2 and H2+

General sets of differential equations have been formulated on the basis sets completeness of the Hilbert space when using operators of the molecular Hamiltonian. In these equations, different relationships between elementary functionals (EF) have been found depending of functions of the overlap, and overlap derivative. Applications of these relationships to simple systems (example: a diatomic molecule with two basis components) were performed. Similarities of early approximations for parametric quantum methods were obtained. The molecular total energy for H2 and H2+ using minimal basis set was found to be as a function of the overlap plus two integration constants. This way, it may be calculated in terms of dissociation energy and equilibrium bond distance of the molecules. Selected trial overlap, as an exponential function, consistently gives values above the exact potential energy curve (PEC) for the ground state. The optimized overlap exponent at each internuclear distance gives better PECs with respect to minimal basis sets. Results suggest an optimal overlap associated to a complete correlated minimal basis set. In particular, a q-exponential function as a model of the overlap gives energy values to a very near exact PEC.

Hydrogen Abstraction of Camphor Catalyzed by Cytochrome P450cam: A QM/MM Study.

A combined quantum mechanics and molecular mechanics (QM/MM, QM = UB3LYP-D3, MM = AMBER) method is used to study the hydrogen abstraction reaction in P450cam catalyzed hydroxylation of camphor in the quartet state. Compared to QM/MM calculations in the literature, this study uses larger basis sets for the most important atoms at the active site and QM/MM Hessian harmonic frequency calculations to determine the standard Gibbs free energy of activation and kinetic isotope effect. The QM/MM covalent boundary is treated with a capping hydrogen atom method, which is simple and robust. An energy barrier of 21.3 kcal/mol and a standard free energy of activation of 16.8 kcal/mol are obtained for this hydrogen abstraction reaction. These values are similar to those reported in the literature, suggesting that when a general protocol is followed, QM/MM results are reproducible. It is found that using a sufficiently large basis set is important to minimize basis set errors.

Self-consistent implementation of meta-GGA functionals for the ONETEP linear-scaling electronic structure package.

Accurate and computationally efficient exchange-correlation functionals are critical to the successful application of linear-scaling density functional theory (DFT). Local and semi-local functionals of the density are naturally compatible with linear-scaling approaches, having a general form which assumes the locality of electronic interactions and which can be efficiently evaluated by numerical quadrature. Presently, the most sophisticated and flexible semi-local functionals are members of the meta-generalized-gradient approximation (meta-GGA) family, and depend upon the kinetic energy density, τ, in addition to the charge density and its gradient. In order to extend the theoretical and computational advantages of τ-dependent meta-GGA functionals to large-scale DFT calculations on thousands of atoms, we have implemented support for τ-dependent meta-GGA functionals in the ONETEP program. In this paper we lay out the theoretical innovations necessary to implement τ-dependent meta-GGA functionals within ONETEP's linear-scaling formalism. We present expressions for the gradient of the τ-dependent exchange-correlation energy, necessary for direct energy minimization. We also derive the forms of the τ-dependent exchange-correlation potential and kinetic energy density in terms of the strictly localized, self-consistently optimized orbitals used by ONETEP. To validate the numerical accuracy of our self-consistent meta-GGA implementation, we performed calculations using the B97M-V and PKZB meta-GGAs on a variety of small molecules. Using only a minimal basis set of self-consistently optimized local orbitals, we obtain energies in excellent agreement with large basis set calculations performed using other codes. Finally, to establish the linear-scaling computational cost and applicability of our approach to large-scale calculations, we present the outcome of self-consistent meta-GGA calculations on amyloid fibrils of increasing size, up to tens of thousands of atoms.

Adapting DFT+U for the Chemically Motivated Correction of Minimal Basis Set Incompleteness.

Recent algorithmic and hardware advances have enabled the application of electronic structure methods to the study of large-scale systems such as proteins with O(10(3)) atoms. Most such methods benefit greatly from the use of reduced basis sets to further enhance their speed, but truly minimal basis sets are well-known to suffer from incompleteness error that gives rise to incorrect descriptions of chemical bonding, preventing minimal basis set use in production calculations. We present a strategy for improving these well-known shortcomings in minimal basis sets by selectively tuning the energetics and bonding of nitrogen and oxygen atoms within proteins and small molecules to reproduce polarized double-ζ basis set geometries at minimal basis set cost. We borrow the well-known +U correction from the density functional theory community normally employed for self-interaction errors and demonstrate its power in the context of correcting basis set incompleteness within a formally self-interaction-free Hartree-Fock framework. We tune the Hubbard U parameters for nitrogen and oxygen atoms on small-molecule tautomers (e.g., cytosine), demonstrate the applicability of the approach on a number of amide-containing molecules (e.g., formamide, alanine tripeptide), and test our strategy on a 10 protein test set where anomalous proton transfer events are reduced by 90% from RHF/STO-3G to RHF/STO-3G+U, bringing the latter into quantitative agreement with RHF/6-31G* results. Although developed with the study of biological molecules in mind, this empirically tuned U approach shows promise as an alternative strategy for correction of basis set incompleteness errors.