Dual Basis Set Research Articles

Coupled Perturbed Approach to Dual Basis Sets for Molecules and Solids. II: Energy and Band Corrections for Periodic Systems

Coupled Perturbed Approach to Dual Basis Sets for Molecules and Solids. II: Energy and Band Corrections for Periodic Systems

Theoretical study of the methylprolithospermate's pKa in aqueous solution

Protonation and deprotonation reactions play an important role in the biological processes. Such processes are fully thermodynamic and occur in complex media where functional groups are surrounded by other chemicals such as molecules, ions, water, etc. In this paper, the compound of interest is the methylprolithospermate (MPL) molecule, which was recently isolated from salvia yunnanensis CH Wright. The intended functional groups for deprotonation are -OH and -COOH. These are the potential groups depicted at four sites located in MPL. Three thermodynamic cycles were required for the estimation of MPL's pKa by the means of DFT at the levels of B3LYP and PW6B95 with the dual basis set 6–311++G(d,p)//6–31+G(d). Cycle 1 is the direct pKa calculation, Cycle 2 is also the direct pKa calculation with proton hydration, and Cycle 3, the proton exchange process with hydroxyl anion hydration. In these cycles, water molecules as, reagents as well as products, were held as monomers in one hand (monomer cycle approaches), and cluster on the other hand (cluster cycle approaches). The isodesmic scheme was used as benchmark method for pKa calculation and yielded: pKa1 = 3.6 ± 0.3, pKa2 = 6.6 ± 0.2, pKa3 = 9.3 ± 0.2, and pKa4 = 14.4. The inclusion of binding energies of MPL anions in the pKa calculation has improved the results. The best cycles depend on the level of waters and the level of acidity. The more the number of waters clustered to MPL anions, the more the deviation of pKa values.

Discontinuous Galerkin method with Voronoi partitioning for quantum simulation of chemistry

To circumvent a potentially dense two-body interaction tensor and obtain lower asymptotic costs for quantum simulations of chemistry, the discontinuous Galerkin (DG) basis set with a rectangular partitioning strategy was recently introduced [McClean et al, New J. Phys. 22, 093015, 2020]. We propose and numerically scrutinize a more general DG basis set construction based on a Voronoi decomposition with respect to the nuclear coordinates. This allows the construction of DG basis sets for arbitrary molecular and crystalline configurations. We here employ the planewave dual basis set as primitive basis set in the supercell model; as a set of grid-based nascent delta functions, the planewave dual functions provide sufficient flexibility for the Voronoi partitioning. The presented implementation of this DG-Voronoi approach is in Python and solely based on PySCF. We numerically investigate the performance, at the mean-field and correlated level of theory for quasi-1D, quasi-2D and fully 3D systems, and exemplify the application to crystalline systems.

Dual Basis Approach for Ab Initio Anharmonic Calculations of Vibrational Spectroscopy: Application to Microsolvated Biomolecules.

A dual electronic basis set approach is introduced for more efficient but accurate calculations of the anharmonic vibrational spectra in the framework of the vibrational self-consistent field (VSCF) theory. In this approach, an accurate basis set is used to compute the vibrational spectra at the harmonic level. The results are used to scale the potential surface from a more modest but much more efficient basis set. The scaling is such that at the harmonic level the new, scaled potential agrees with one of the accurate basis sets. The approach is tested in the application of the microsolvated, protected amino acid Ac-Phe-OMe, using the scaled anharmonic hybrid potential in the VSCF and VSCF-PT2 algorithms. The hybrid potential method yields results that are in good accord with the experiment and very close to those obtained in calculations with the high-level, very costly potential from the large basis set. At the same time, the hybrid potential calculations are considerably less expensive. The results of the hybrid calculations are much more accurate than those computed from the potential surface corresponding to the modest basis set. The results are very encouraging for using the hybrid potential method for inexpensive yet sufficiently accurate anharmonic calculations for the spectra of large biomolecules.

Coupled Perturbation Theory Approach to Dual Basis Sets for Molecules and Solids. 1. General Theory and Application to Molecules.

We present a new coupled Hartree-Fock(HF)/Kohn-Sham DFT perturbation method that accounts for the effect of enlarging the basis set in electronic structure calculations. In contrast with previous approaches, our dual basis set treatment yields not only a correction for the total energy but also for the orbital eigenvalues and density. The zeroth order solution is obtained from the projection of the small basis set coefficients. Diagonalization of the full Fock matrix in the large basis set is avoided. In this first paper of a series, we develop the theoretical foundations of our approach for molecules, including the coupled-perturbed equations through second order and the energy expressions through fourth order-as our method complies with Wigner's 2n + 1 rule. The first-order perturbation equation turns out to be uncoupled, and odd-order terms in the energy expansion vanish. In calculations on simple molecules, our method recovers over 93% (84%) of the missing DFT(HF) energy when going from the cc-pVDZ to the aug-cc-pVDZ basis, and over about 95% in all cases if an energy extrapolation formula is used. Mulliken charges, the orbital eigenvalue spectrum, and HOMO-LUMO gaps of the large basis are well reproduced. Charge density maps show that the differences between the perturbatively corrected density and the reference nearly vanish through second-order.

Dual Basis Set Approach for Density Functional and Wave Function Embedding Schemes

A dual basis (DB) approach is proposed which is suitable for the reduction of the computational expenses of the Hartree-Fock, Kohn-Sham, and wave function-based correlation methods. The approach is closely related to the DB approximation of Head-Gordon and co-workers [ J. Chem. Phys. 2006 , 125 , 074108 ] but specifically designed for embedding calculations. The new approach is applied to our variant of the projector-based embedding theory utilizing the Huzinaga-equation, multilevel local correlation methods, and combined density functional-multilevel local correlation approximations. The performance of the resulting DB density functional and wave function embedding methods is evaluated in extensive benchmark calculations and also compared to that of the corresponding embedding schemes exploiting the mixed-basis approximation. Our results show that, with an appropriate combination of basis sets, the DB approach significantly speeds up the embedding calculations, and, for chemical processes where the electronic structure considerably changes, it is clearly superior to the mixed-basis approximation. The results also demonstrate that the DB approach, if integrated with the mixed-basis approximation, efficiently eliminates the major weakness of the latter, and the combination of the DB and mixed-basis schemes is the most efficient strategy to accelerate embedding calculations.

Magnesium-Ion Distribution and Dynamics in Magnesium Zirconium Orthophosphate

The magnesium secondary battery has drawn much attention as a post lithium-ion secondary battery at recent years because of the higher theoretical volumetric energy density. Similarly to the lithium-ion battery, inorganic oxides such as MgCo2O4 [1] and MgFeSiO4 [2] can be regarded as one of the most promising candidates for a cathode material of the magnesium secondary battery. Due to a strong electrostatic interaction between Mg2+ and surrounding oxide anions, however, Mg2+ diffusion is deteriorated significantly in oxides and thus the battery performance is very low generally at room temperature. In order to overcome the problem, we have to carry out a systematic study on Mg2+ diffusion in solids and gain deeper understanding on Mg2+ dynamics. From such background, this work focuses on MgZr4(PO4)6-based Mg-ion conductors [3], and investigated Mg2+ behaviors in the crystal by means of a combination of the density functional theory (DFT) and the reverse Monte Carlo (RMC) modelling using synchrotron X-ray total scatterings. We synthesized Mg1-2x(Zr1-xNbx)4(PO4)6 with a conventional solid-state reaction method using MgHPO4·3H2O, ZrO(NO3)2·2H2O, Nb2O5 and NH4H2PO4 as starting materials. In the sintering process, we performed the spark plasma sintering (SPS: LABOX-315, Sinter Land Inc.) in order to prepare dense ceramics of the samples for electrical conductivity measurements. For refinements of crystal structures (average structures) of the samples, synchrotron X-ray Bragg profiles were recorded at BL19B2 installed at SPring-8, Japan, and then analyzed by the Rietveld and maximum-entropy method (MEM) techniques. From the refined unit cells, we constructed super cells comprised of 276 atoms or above, and Mg-ion distributions were determined experimentally by the RMC simulations for polycrystalline materials [4, 5] using the Bragg profiles and X-ray structure factors S(Q) simultaneously. The S(Q) were measured with BL04B2 at SPring-8, Japan, and then degraded by convolutions considering the simulation box sizes. As a theoretical approach, we performed the DFT calculations and the DFT-based molecular dynamic (MD) simulations with the CP2K program which combines the localized Gaussian basis set and plane waves for a dual GPW basis set. As the exchange-correlation energy functional, the generalized gradient approximation of PBE was utilized. In the DFT-based MD simulation, the NVT ensemble was adopted, and the temperature was controlled with a CSVR thermostat. Initial cells for these calculations were constructed on the basis of the average structures or the atomic-configuration snapshots obtained by the RMC modelling. From preliminary laboratorial X-ray diffraction measurements, it is confirmed that Mg1-2x(Zr1-xNbx)4(PO4)6 can be synthesized successfully at least within the Nb-content range of x=0~0.25. The Rietveld analysis using synchrotron X-ray data reveals that the Nb-substituted samples have a monoclinic average structure with a space group of P21/n which is the same as MgZr4(PO4)6. By using the refined unit cell, we constructed a super cell with 276 atoms and then performed DFT-base MD simulation to visualize Mg2+ diffusion. As a result, it is indicated that Mg2+ at an experimentally-determined crystallographic site with five-fold coordination migrates via another site. It is also found that a triangle formed by three oxide ions becomes a bottle neck for the migration. Such a diffusion pathway is supported by a bond-valence-sum (BVS) mapping using an atomic configuration relaxed energetically by the DFT. Acknowledgment This work was financially supported in part by the ALCA-SPRING project.

Third-Order Incremental Dual-Basis Set Zero-Buffer Approach for Large High-Spin Open-Shell Systems.

The third-order incremental dual-basis set zero-buffer approach (inc3-db-B0) is an efficient, accurate, and black-box quantum chemical method for obtaining correlation energies of large systems, and it has been successfully applied to many real chemical problems. In this work, we extend this approach to high-spin open-shell systems. In the open-shell approach, we will first decompose the occupied orbitals of a system into several domains by a K-means clustering algorithm. The essential part is that we preserve the active (singly occupied) orbitals in all the calculations of the domain correlation energies. The duplicated contributions of the active orbitals to the correlation energy are subtracted from the incremental expansion. All techniques of truncating the virtual space such as the B0 approximation can be applied. This open-shell inc3-db-B0 approach is combined with the CCSD and CCSD(T) methods and applied to the computations of a singlet-triplet gap and an electron detachment process. Our approach exhibits an accuracy better than 0.6 kcal/mol or 0.3 eV compared with the standard implementation, while it saves a large amount of the computational time and can be efficiently parallelized.

Variational nature of the frozen density energy in density-based energy decomposition analysis and its application to torsional potentials.

The density-based energy decomposition analysis (DEDA) is the first of its kind to calculate the frozen density energy variationally. Defined with the constrained search formulation of density functional theory, the frozen density energy is optimized in practice using the Wu-Yang (WY) method for constrained minimizations. This variational nature of the frozen density energy, a possible reason behind some novel findings of DEDA, will be fully investigated in this work. In particular, we systematically study the dual basis set dependence in WY: the potential basis set used to expand the Lagrangian multiplier function and the regular orbital basis set. We explain how the convergence progresses differently on these basis sets and how an apparent basis-set independence is achieved. We then explore a new development of DEDA in frozen energy calculations of the ethane molecule, focusing on the internal rotation around the carbon-carbon bond and the energy differences between staggered and eclipsed conformations. We argue that the frozen density energy change at fixed bond lengths and bond angles is purely steric effects. Our results show that the frozen density energy profile follows closely that of the total energy when the dihedral angle is the only varying geometry parameter. We can further analyze the contributions from electrostatics and Pauli repulsions. These results lead to a meaningful DEDA of the torsional potential in ethane.

Approaching the complete basis set limit of CCSD(T) for large systems by the third-order incremental dual-basis set zero-buffer F12 method.

The third-order incremental dual-basis set zero-buffer approach was combined with CCSD(T)-F12x (x = a, b) theory to develop a new approach, i.e., the inc3-db-B0-CCSD(T)-F12 method, which can be applied as a black-box procedure to efficiently obtain the near complete basis set (CBS) limit of the CCSD(T) energies also for large systems. We tested this method for several cases of different chemical nature: four complexes taken from the standard benchmark sets S66 and X40, the energy difference between isomers of water hexamer and the rotation barrier of biphenyl. The results show that our method has an error relative to the best estimation of CBS energy of only 0.2 kcal/mol or less. By parallelization, our method can accomplish the CCSD(T)-F12 calculations of about 60 correlated electrons and 800 basis functions in only several days, which by standard implementation are impossible for ordinary hardware. We conclude that the inc3-db-B0-CCSD(T)-F12a/AVTZ method, which is of CCSD(T)/AV5Z quality, is close to the limit of accuracy that one can achieve for large systems currently.

Third-Order Incremental Dual-Basis Set Zero-Buffer Approach: An Accurate and Efficient Way To Obtain CCSD and CCSD(T) Energies.

An efficient way to obtain accurate CCSD and CCSD(T) energies for large systems, i.e., the third-order incremental dual-basis set zero-buffer approach (inc3-db-B0), has been developed and tested. This approach combines the powerful incremental scheme with the dual-basis set method, and along with the new proposed K-means clustering (KM) method and zero-buffer (B0) approximation, can obtain very accurate absolute and relative energies efficiently. We tested the approach for 10 systems of different chemical nature, i.e., intermolecular interactions including hydrogen bonding, dispersion interaction, and halogen bonding; an intramolecular rearrangement reaction; aliphatic and conjugated hydrocarbon chains; three compact covalent molecules; and a water cluster. The results show that the errors for relative energies are <1.94 kJ/mol (or 0.46 kcal/mol), for absolute energies of <0.0026 hartree. By parallelization, our approach can be applied to molecules of more than 30 atoms and more than 100 correlated electrons with high-quality basis set such as cc-pVDZ or cc-pVTZ, saving computational cost by a factor of more than 10-20, compared to traditional implementation. The physical reasons of the success of the inc3-db-B0 approach are also analyzed.

Computational study of molecular properties with dual basis sets

We have studied the performance of dual basis (DB) sets for the evaluation of molecular properties via second order Møller-Plesset perturbation theory (MP2). In addition to savings derived from using a trimmed basis set for the underlying Hartree-Fock (HF) calculation, we pursued a systematic truncation of the virtual subspace for further reductions in computational overhead during the post-HF step. Calculated total energies and molecular properties within the DB framework without virtual space truncation are generally in excellent agreement with full basis calculations. When aug-cc-pV5Z is used as the parent basis, mean absolute error for DB-HF (DB-MP2) total energies of molecules within our test set is 9.7 × 10(-5) au (8.0 × 10(-5) au) while mean absolute relative errors for static electrical response properties like dipole moments, isotropic dipole polarizabilities and polarizability anisotropies are 0.15% (0.14%), 0.56% (0.72%), and 0.76% (0.83%) respectively. When DB is coupled with virtual space truncation at the MP2 level, the corresponding errors are larger but still within 2% of full basis values.

Approaching the theoretical limit in periodic local MP2 calculations with atomic-orbital basis sets: The case of LiH

The atomic orbital basis set limit is approached in periodic correlated calculations for solid LiH. The valence correlation energy is evaluated at the level of the local periodic second order Møller-Plesset perturbation theory (MP2), using basis sets of progressively increasing size, and also employing "bond"-centered basis functions in addition to the standard atom-centered ones. Extended basis sets, which contain linear dependencies, are processed only at the MP2 stage via a dual basis set scheme. The local approximation (domain) error has been consistently eliminated by expanding the orbital excitation domains. As a final result, it is demonstrated that the complete basis set limit can be reached for both HF and local MP2 periodic calculations, and a general scheme is outlined for the definition of high-quality atomic-orbital basis sets for solids.

Solenoidal bases for numerical studies of transition in pipe flow

In this paper, solenoidal basis functions are employed in numerical studies of transition in incompressible pipe flow. The bases are formulated in terms of Legendre polynomials, which are more favorable both for the functional form of the basis functions and for the inner product integrals arising in the Galerkin projection scheme. The projection is performed onto the dual solenoidal basis set to eliminate the pressure gradient term from the governing equations. This simplifies the numerical approach to the problem and the resulting scheme is used to study the nonlinear stability of pipe flow. Some preliminary results are compared with those found in the literature.

Ab Initio Molecular Dynamics with Dual Basis Set Methods

On-the-fly, ab initio classical molecular dynamics are demonstrated with an underlying dual basis set potential energy surface. Dual-basis self-consistent field (Hartree-Fock and density functional theory) and resolution-of-the-identity second-order Møller-Plesset perturbation theory (RI-MP2) dynamics are tested for small systems, including the water dimer. The resulting dynamics are shown to be faithful representations of their single-basis analogues for individual trajectories, as well as vibrational spectra. Computational cost savings of 58% are demonstrated for SCF methods, even relative to Fock-extrapolated dynamics, and savings are further increased to 71% with RI-MP2. Notably, these timings outperform an idealized estimate of extended-Lagrangian molecular dynamics. The method is subsequently demonstrated on the vibrational absorption spectrum of two NO(+)(H₂O)₃ isomers and is shown to recover the significant width of the shared-proton bands observed experimentally.

Carbohydrate-Aromatic Interactions: The Role of Curvature on XH···π Interactions

The interaction between the fragment of carbon nanotube (CNT) and carbohydrates has been investigated using MP2 and M05-2X methods using various basis sets in gas phase. Three carbohydrates, viz., beta-d-glucose, beta-d-galactose, and beta-d-xylose with different degree of hydrophobic nature have been selected for this investigation. With a view to assess the effect of curvature on the interaction between the carbohydrates and CNT, calculations on intermolecular complexes comprising of coronene (COR) and carbohydrates have also been carried out in gas phase. Results obtained from electronic structure calculations combined with the Bader's electron density analysis reveal that CH...pi interaction is the predominant one in the stabilization of the carbohydrate-CNT and carbohydrate-COR complexes. Furthermore, the importance of OH...pi and lone pair...pi (lp...pi) interactions are also evident from the results. The calculated BEs for the various carbohydrate-CNT and carbohydrate-COR complexes at M05-2X with dual basis set [aug-cc-pVTZ for carbohydrate + cc-pVTZ for both CNT and COR] vary from -2.52 to -5.14 and from -4.14 to -8.04 kcal/mol, respectively. The corresponding BEs obtained from MP2/6-311++G(d,p)//M05-2X/6-31+G(d,p) level of calculation range from -4.92 to -9.93 and from -6.75 to -12.53 kcal/mol. Close scrutiny of the energetics of all the complexes elucidate that the electron correlation energy (dispersion energy) significantly contribute to the stability of these complexes. It is found from the analysis of geometrical parameters and BEs that the interplay of orientation of the X-H (X = C and O) bond to the pi-surface is crucial for the recognition and further stabilization. Molecular electrostatic potential (MESP) isosurfaces of curved and planar surfaces have clearly provided the difference between the pi-electron distributions. Evidences form the energy decomposition analysis elicit that the dispersive interaction plays a significant role in the overall stabilization of the complexes. And, it is possible to observe the delicate balance between the electrostatic interaction and the exchange-repulsion energy.

Efficient Calculation of Heats of Formation

An efficient procedure has been devised for calculating heats of formation of uncharged, closed-shell molecules comprising H, C, N, O, F, S, Cl, and Br. Known as T1, it follows the G3(MP2) recipe, by substituting an HF/6-31G* for the MP2/6-31G* geometry, eliminating both the HF/6-31G* frequency and QCISD(T)/6-31G* energy and approximating the MP2/G3MP2large energy using dual basis set RI-MP2 techniques. Taken together, these changes reduce computation time by 2-3 orders of magnitude. Atom counts, Mulliken bond orders, and HF/6-31G* and RI-MP2 energies are introduced as variables in a linear regression fit to a set of 1126 G3(MP2) heats of formation. The T1 procedure reproduces these values with mean absolute and rms errors of 1.8 and 2.5 kJ/mol, respectively. It reproduces experimental heats of formation for a set of 1805 diverse organic molecules from the NIST thermochemical database with mean absolute and rms errors of 8.5 and 11.5 kJ/mol, respectively. Heats of formation of flexible molecules have been approximated by the heats of formation of their lowest-energy conformer as given by the T1 recipe. This has been identified by examining all conformers for molecules with fewer than 100 conformers and by examining a random sample of 100 conformers for molecules with more than 100 conformers. While this approximation necessarily yields heats of formation that are too negative, the error for typical organic molecules with less than 10 degrees of conformational freedom (several thousand conformers) is <2-3 kJ/mol. T1 heats of formation have been used to calculate energy differences for a variety of structural, positional, and stereoisomers, as well as energy differences between conformers in a variety of simple acyclic and cyclic molecules for which reliable experimental data are available. In terms of both overall error and errors for individual systems, T1 provides a better account of the experimental thermochemistry than any practical quantum chemical method that we have previously examined. A database of approximately 40,000 T1 calculations for both rigid and flexible organic molecules has been produced and is available as part of the Spartan Molecular Database (SMD) in the current version of the Spartan electronic structure program (Spartan'08). (A subset of approximately 5000 molecules is provided as part of the standard release, and the full T1 database can be licensed.). This collection differs from the other components of SMD in that the lowest-energy conformation for each molecule has been assigned using a high-level quantum chemical method and not molecular mechanics. Thus, it is not only a source of "high-quality" calculated heats of formation for organic molecules but also a source of conformational preferences.

Molecular properties with dual basis set methods

The dual basis set approach has proven to be very successful for accurately estimating total energies with large basis sets. This study extends the applications of this technique to the calculation of molecular properties, including energy derivatives with respect to nuclear positions and to an external electric field. All energy derivatives have been calculated numerically via finite-differences. Molecular gradients and Hessians as well as dipole moments and polarizabilites have been calculated at the HF and MP2 levels using two alternative versions of the dual basis set method. The accuracy of these approaches is discussed in the context of quality of basis sets used in calculations. It is shown that even quite poor results obtained with the 6-311G basis set are significantly improved in dual basis set calculations with the 6-311G(d,p) and 6-311G(3df,3dp) basis sets.

An Extended Group Function Model for Intermolecular Interactions

An extended group function model is introduced for the calculation of intermolecular interactions. The model is formulated within the framework of the energy incremental scheme. In the calculation of intra- and intersystem energies, model systems are introduced. To each subsystem is associated a set of partner subsystems defined by a vicinity criterion. In the independent calculation of intra- and intersystem energies, calculations are performed on model systems defined by the subsystems considered and their partner subsystems. To further reduce computation time, dual basis sets are introduced. A small and a large basis set are associated with each subsystem. For partner subsystems in a model system, the small basis set is adopted. Test calculations are performed on helium atoms in one- and two-dimensional lattices.

Fast Hartree–Fock theory using local density fitting approximations

Density fitting approximations are applied to generate the Fock matrix in Hartree–Fock calculations. By localizing the orbitals in each iteration and performing separate fits for each orbital the scaling of the computational effort for the exchange can be reduced to . We also use the Poisson method to replace almost all Coulomb integrals with simple overlaps, an efficient alternative to diagonalization, and dual basis sets such that the Hartree–Fock calculation is performed in a smaller basis than the subsequent treatment of electron correlation. The accuracy and efficiency of the method is demonstrated in calculations with almost 4000 basis functions. The errors introduced by the local approximations on HF and MP2 energies are small compared to those that arise from the density fitting, and the fitting errors themselves (typically 1–10 microhartree per atom) are very small compared, for example, to the effect of basis set variations.