Basis Set Truncation Research Articles

Duo: A general program for calculating spectra of diatomic molecules

Duo is a general, user-friendly program for computing rotational, rovibrational and rovibronic spectra of diatomic molecules. Duo solves the Schrödinger equation for the motion of the nuclei not only for the simple case of uncoupled, isolated electronic states (typical for the ground state of closed-shell diatomics) but also for the general case of an arbitrary number and type of couplings between electronic states (typical for open-shell diatomics and excited states). Possible couplings include spin–orbit, angular momenta, spin-rotational and spin–spin. Corrections due to non-adiabatic effects can be accounted for by introducing the relevant couplings using so-called Born–Oppenheimer breakdown curves.Duo requires user-specified potential energy curves and, if relevant, dipole moment, coupling and correction curves. From these it computes energy levels, line positions and line intensities. Several analytic forms plus interpolation and extrapolation options are available for representation of the curves. Duo can refine potential energy and coupling curves to best reproduce reference data such as experimental energy levels or line positions. Duo is provided as a Fortran 2003 program and has been tested under a variety of operating systems. Program summaryProgram title:DuoCatalogue identifier: AEZJ_v1_0rogram summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 187443No. of bytes in distributed program, including test data, etc.: 6968371Distribution format: tar.gzProgramming language: Fortran 2003.Computer: Any personal computer.Operating system: Linux, Windows, Mac OS.Has the code been vectorized or parallelized?: ParallelizedRAM: Case dependent, typically < 10 MBClassification: 4.3, 4.9, 16.2, 16.3.Nature of problem: Solving the Schrödinger equation for the nuclear motion of a diatomic molecule with an arbitrary number and type of couplings between electronic states.Solution method: Solution of the uncoupled problem first, then basis set truncation and solution of the coupled problem. A line list can be computed if a dipole moment function is provided. The potential energy and other curves can be empirically refined by fitting to experimental energies or frequencies, when provided.Restrictions: The current version is restricted to bound states of the system.Unusual features: User supplied curves for all objects (potential energies, spin–orbit and other couplings, dipole moment etc.) as analytic functions or on a grid is a program requirement.Running time: Case dependent. The test runs provided take seconds or a few minutes on a normal PC.

Ab initio investigation of sulfur monofluoride and its singly charged cation and anion in their ground electronic state**Project supported by the National Natural Science Foundation of China (Grant Nos. 11304023 and 11447172), the Young and Middle-Aged Talent of Education Burea of Hubei Province, China (Grant No. Q20151307), and the Yangtze Youth Talents Fund of Yangtze University, China (Grant No. 2015cqr21).

The SF radical and its singly charged cation and anion, SF+ and SF−, have been investigated on the MRCI/aug-cc-pVXZ (X = Q, 5, 6) levels of theory with Davidson correction. Both the core–valence correlation and the relativistic effect are considered. The extrapolating to the complete basis set (CBS) limit is adopted to remove the basis set truncation error. Geometrical parameters, potential energy curves (PECs), vibrational energy levels, spectroscopic constants, ionization potentials, and electron affinities of the ground electronic state for all these species are obtained. The information with respect to molecular characteristics of the SFn (n = −1, 0, +1) systems derived in this work will help to extend our knowledge and to guide further experimental or theoretical researches.

Application of a convergent, composite coupled cluster approach to bound state, adiabatic electron affinities in atoms and small molecules.

Benchmark quality adiabatic electron affinities for a collection of atoms and small molecules were obtained with the Feller-Peterson-Dixon composite coupled cluster theory method. Prior applications of this method demonstrated its ability to accurately predict atomization energies/heats of formation for more than 170 molecules. In the current work, the 1-particle expansion involved very large correlation consistent basis sets, ranging up to aug-cc-pV9Z (aug-cc-pV10Z for H and H2), with the goal of minimizing the residual basis set truncation error that must otherwise be approximated with extrapolation formulas. The n-particle expansion begins with coupled cluster calculations through iterative single and double excitations plus a quasiperturbative treatment of "connected" triple excitations (CCSD(T)) pushed to the complete basis set limit followed by CCSDT, CCSDTQ, or CCSDTQ5 corrections. Due to the small size of the systems examined here, it was possible in many cases to extend the n-particle expansion to the full configuration interaction wave function limit. Additional, smaller corrections associated with core/valence correlation, scalar relativity, anharmonic zero point vibrational energies, and non-adiabatic effects were also included. The overall root mean square (RMS) deviation was 0.005 eV (0.12 kcal/mol). This level of agreement was comparable to what was found with molecular heats of formation. A 95% confidence level corresponds to roughly twice the RMS value or 0.01 eV. While the atomic electron affinities are known experimentally to high accuracy, the molecular values are less certain. This contributes to the difficulty of gauging the accuracy of the theoretical results. A limited number of electron affinities were determined with the explicitly correlated CCSD(T)-F12b method. After extending the VnZ-F12 orbital basis sets with additional diffuse functions, the F12b method was found to accurately reproduce the best F/F(-) value obtained with standard methods, but it underestimated the CH3/CH3 (-) value by 0.01 eV.

Single and double valence configuration interactions for recovering the exponential decay law while tunneling through a molecular wire

The exponential decay of the electronic transmission through a molecular wire with its length is calculated using a configuration interaction elastic scattering quantum chemistry (CI-ESQC) theory [, ]. In the HOMO–LUMO gap and in a one-electron approximation, this decay is exponential since the scattering matrix comes from a product of spatial propagators along the wire. In a valence SD–CI (single and double-configurations interaction) description, such a product does not exist. An effective one was numerically obtained from the CI-ESQC scattering matrix. Fluctuations over the effective CI-exponential decay come from the truncation of the full CI basis set and also from many-body exchange-correlation effects along the molecular wire.

An ab initio investigation on the ground electronic state of chlorine monoxide and its singly charged cation and anion

The MRCI method has been utilized to calculate the equilibrium structure of the ground electronic state for the ClO radical and its singly charged cation and anion, ClO+ and ClO−. The augmented correlation-consistent basis sets up through sextuple-zeta quality are used to derive equilibrium structural parameters, potential energy curves and spectroscopic constants of the systems. Two extrapolating schemes enable us to remove the basis set truncation error and to estimate the complete basis set limit. Corrections of core-valence correlation and relativistic effect are included in our calculations. By solving the radial Schrödinger equation of nuclear motion, the vibrational energy levels as well as rotational and centrifugal distortion constants of the ground electronic states for the three species are obtained. Ionization potentials and the electron affinities are also obtained on the RCCSD(T)/AV6Z level. The knowledge extracted from this work are anticipated to extend our understanding on molecular characteristics of the ClOn (n=−1, 0, +1) systems and can serve as guidelines for further experimental or theoretical researches.

Large-scale density functional theory simulation of inorganic nanotubes: a case study on Imogolite nanotubes

The high experimental control over inorganic Imogolite-like open-ended nanotubes (Imo-NTs) composition, dimensions and monodispersity together with the potentially huge range of tuneable properties that can be introduced by chemical functionalisation and doping make Imo-NTs appealing substrates for nanotechnology, as artificial ion-channels and in chemical separation. Investigation of Imo-NTs electronic and spectroscopic properties has so far been hampered by the large size of the systems repeat unit (+300 atoms), which pose severe challenges and accuracy-viability compromises for standard plane-wave (fixed atomic basis set) density functional theory (DFT) simulations. These challenges can, however, be met by linear-scaling DFT (LS-DFT) approaches based on in situ optimisation of minimal basis sets. Here, we illustrate the applicability of LS-DFT to Imo-NTs by providing structural and electronic characterisation of periodic and finite models of aluminosilicate (AlSi) and methylated-AlSi Imo-NTs. It is shown that adoption of moderate kinetic energy cutoff (1000 eV) and basis set truncation radius (8 Bohr) leads to optimal accuracy-viability compromised for geometrical optimisation of Imo-NTs. These results should be useful for future LS-DFT investigation of Imo-NTs and other AlSi-based functional materials.

Green's function multiple-scattering theory with a truncated basis set: An augmented-KKR formalism

Korringa-Kohn-Rostoker (KKR) Green's function, multiple-scattering theory is an efficient site-centered, electronic-structure technique for addressing an assembly of $N$ scatterers. Wave-functions are expanded in a spherical-wave basis on each scattering center and indexed up to a maximum orbital and azimuthal number $L_{max}=(l,m)_{max}$, while scattering matrices, which determine spectral properties, are truncated at $L_{tr}=(l,m)_{tr}$ where phase shifts $\delta_{l>l_{tr}}$ are negligible. Historically, $L_{max}$ is set equal to $L_{tr}$; however, a more proper procedure retains free-electron and single-site contributions for $L_{max}>L_{tr}$ with $\delta_{l>l_{tr}}$ set to zero [Zhang and Butler, Phys. Rev. B {\bf 46}, 7433]. We present a numerically efficient and accurate \emph{augmented}-KKR Green's function formalism that solves the KKR secular equations by matrix inversion [$\mathcal{R}^3$ process with rank $N(l_{tr}+1)^2$] and includes higher-order $L$ contributions via linear algebra [$\mathcal{R}^2$ process with rank $N(l_{max}+1)^2$]. Augmented-KKR yields properly normalized wave-functions, numerically cheaper basis-set convergence, and a total charge density and electron count that agrees with Lloyd's formula. For fcc Cu, bcc Fe and L$1_0$ CoPt, we present the formalism and numerical results for accuracy and for the convergence of the total energies, Fermi energies, and magnetic moments versus $L_{max}$ for a given $L_{tr}$.

The Coulson–Fischer wave functions for the ground and excited states of H2 and HeH+: parametrisation using distributed Gaussian basis sets

Coulson–Fischer wave functions are used to determine complete potential energy curves for the X1Σ+g ground state of the H2 molecule and for the first excited state (EF1Σ+g) having the same symmetry. The Coulson–Fischer orbitals are parametrised by expansion in distributed Gaussian basis sets of s-type functions. The exponents of the Gaussian functions are generated by using an even-tempered prescription. An anharmonic model is used to locate the basis functions along the internuclear axis. Sequences of basis sets are used to reduce the basis set truncation error to the sub-μhartree level. Equations are derived to determine the generalised Coulson–Fischer orbitals for the excited state which ensure that orthogonality conditions with respect to the ground state are satisfied. Potential energy curves are also calculated using smaller basis sets for which both exponents and basis function positions are determined by invoking the variation principle. The results of calculations for the isoelectronic heteronuclear HeH+ ion are also reported. The potential energy curves obtained in the present study are compared with literature values for both H2 and HeH+.

Molecule-Optimized Basis Sets and Hamiltonians for Accelerated Electronic Structure Calculations of Atoms and Molecules

Molecule-optimized basis sets, based on approximate natural orbitals, are developed for accelerating the convergence of quantum calculations with strongly correlated (multireferenced) electrons. We use a low-cost approximate solution of the anti-Hermitian contracted Schrödinger equation (ACSE) for the one- and two-electron reduced density matrices (RDMs) to generate an approximate set of natural orbitals for strongly correlated quantum systems. The natural-orbital basis set is truncated to generate a molecule-optimized basis set whose rank matches that of a standard correlation-consistent basis set optimized for the atoms. We show that basis-set truncation by approximate natural orbitals can be viewed as a one-electron unitary transformation of the Hamiltonian operator and suggest an extension of approximate natural-orbital truncations through two-electron unitary transformations of the Hamiltonian operator, such as those employed in the solution of the ACSE. The molecule-optimized basis set from the ACSE improves the accuracy of the equivalent standard atom-optimized basis set at little additional computational cost. We illustrate the method with the potential energy curves of hydrogen fluoride and diatomic nitrogen. Relative to the hydrogen fluoride potential energy curve from the ACSE in a polarized triple-ζ basis set, the ACSE curve in a molecule-optimized basis set, equivalent in size to a polarized double-ζ basis, has a nonparallelity error of 0.0154 au, which is significantly better than the nonparallelity error of 0.0252 au from the polarized double-ζ basis set.

Diffraction of helium on MgO(100) surface calculated from first-principles.

In this work we simulate the diffraction peak intensities of He beams scattered on the MgO(100) surface from first principles. It turns out that diffraction peak intensities are extremely sensitive to the quality of the potential describing the He-MgO surface interaction. Achieving the required accuracy in first principles calculations is very challenging indeed. The present work describes a first principles protocol able to achieve very high accuracy for reasonable computational cost. This method is based on periodic local second-order Møller-Plesset perturbation theory where systematic corrections for basis set truncation and for high-order electronic correlation are introduced using coupled cluster calculations on finite model systems mimicking the target system. For the He-MgO system the requirements with respect to the level of theory are very high; it turns out that contributions from connected quadruple excitations are non-negligible. Here we demonstrate that using this protocol, it is possible to reach the accuracy in the He-MgO potential that is required to predict the observed He diffraction peak intensities.

Accurate basis set truncation for wavefunction embedding

Density functional theory (DFT) provides a formally exact framework for performing embedded subsystem electronic structure calculations, including DFT-in-DFT and wavefunction theory-in-DFT descriptions. In the interest of efficiency, it is desirable to truncate the atomic orbital basis set in which the subsystem calculation is performed, thus avoiding high-order scaling with respect to the size of the MO virtual space. In this study, we extend a recently introduced projection-based embedding method [F. R. Manby, M. Stella, J. D. Goodpaster, and T. F. Miller III, J. Chem. Theory Comput. 8, 2564 (2012)] to allow for the systematic and accurate truncation of the embedded subsystem basis set. The approach is applied to both covalently and non-covalently bound test cases, including water clusters and polypeptide chains, and it is demonstrated that errors associated with basis set truncation are controllable to well within chemical accuracy. Furthermore, we show that this approach allows for switching between accurate projection-based embedding and DFT embedding with approximate kinetic energy (KE) functionals; in this sense, the approach provides a means of systematically improving upon the use of approximate KE functionals in DFT embedding.

Accurate non-covalent interactions with basis-set corrections from interference-corrected perturbation theory: comparison with the S22B database

Non-covalent interactions can usually be described accurately at the coupled-cluster level of theory using single, double and triple excitations, the latter within a perturbation-theory framework. However, since the computations with the corresponding coupled-cluster model, CCSD(T), are computationally very demanding, they can often not be carried out in large enough basis sets. This leads to a basis-set truncation error. To correct this error, a basis-set correction can be computed at the level of second-order Møller–Plesset (MP2) perturbation theory in the limit of a complete basis set, but such a correction tends to overestimate the magnitude of the basis-set truncation error. In the present work, we suggest to damp the basis-set correction obtained at the complete-basis-set MP2 level using interference factors for individual occupied orbital pairs. The approach is applied to the non-covalent interactions of the S22B database, where the interference correction turns out to be very small.

Accurate Noncovalent Interaction Energies Using Truncated Basis Sets Based on Frozen Natural Orbitals.

We assess the accuracy of basis set truncations based on natural orbitals determined by second-order perturbation theory for computing noncovalent interaction energies with coupled cluster through perturbative triples [CCSD(T)]. We consider two methods for truncation: (i) the usual frozen natural orbital approach (FNO) in which the basis set truncation occurs before the iterative CCSD computation [FNO CCSD(T)] and (ii) an approach in which the truncation occurs only for the perturbative triples contribution [CCSD+FNO(T)]. The errors incurred are comparable for both methods and are small enough for the methods to be used for benchmark-quality studies of noncovalent interactions. For the FNO CCSD(T) method with a modest natural orbital occupation tolerance of 10(-5), the mean absolute error in the interaction energies for the S22 data set in an aug-cc-pVDZ basis set is only 0.012 kcal mol(-1) versus canonical CCSD(T) values. The same method exhibits a mean absolute error of 0.020 kcal mol(-1) for the S11 data set in the aug-cc-pVTZ basis set versus canonical CCSD(T) values.

Convergence of many-body wave-function expansions using a plane-wave basis: From homogeneous electron gas to solid state systems

Using the finite simulation-cell homogeneous electron gas (HEG) as a model, we investigate the convergence of the correlation energy to the complete-basis-set (CBS) limit in methods utilizing plane-wave wave-function expansions. Simple analytic and numerical results from second-order M\o{}ller-Plesset theory (MP2) suggest a $1/M$ decay of the basis-set incompleteness error where $M$ is the number of plane waves used in the calculation, allowing for straightforward extrapolation to the CBS limit. As we shall show, the choice of basis-set truncation when constructing many-electron wave functions is far from obvious, and here we propose several alternatives based on the momentum transfer vector, which greatly improve the rate of convergence. This is demonstrated for a variety of wave-function methods, from MP2 to coupled-cluster doubles theory and the random-phase approximation plus second-order screened exchange. Finite basis-set energies are presented for these methods and compared with exact benchmarks. A transformation can map the orbitals of a general solid state system onto the HEG plane-wave basis and thereby allow application of these methods to more realistic physical problems. We demonstrate this explicitly for solid and molecular lithium hydride.

On the gauge invariance of nonperturbative electronic dynamics using the time-dependent Hartree-Fock and time-dependent Kohn-Sham

Nonperturbative electronic dynamics using the time-dependent Hartree-Fock (TDHF) and time-dependent Kohn-Sham (TDKS) theories with the adiabatic approximation is a powerful tool in obtaining insights into the interaction between a many-electron system and an external electromagnetic field. In practical applications of TDHF/TDKS using a truncated basis set, the electronic dynamics and molecular properties become gauge-dependent. Numerical simulations are carried out in the length gauge and velocity gauge to verify the extent of gauge-dependence using incomplete basis sets. Electronic dynamics of two many-electron systems, a helium atom and a carbon monoxide molecule in high-intensity linearly polarized radiation fields are performed using the TDHF and TDKS with two selected adiabatic exchange-correlation (xc) functionals. The time evolution of the expectation values of the dipole moment and harmonic spectra are calculated in the two gauges, and the basis set dependence on the gauge-invariance of these properties is investigated.

Simulating strongly correlated multiparticle systems in a truncated Hilbert space

Representing a strongly interacting multi-particle wave function in a finite product basis leads to errors. Simple rescaling of the contact interaction can preserve the low-lying energy spectrum and long-wavelength structure of wave functions in one-dimensional systems and thus correct for the basis set truncation error. The analytic form of the rescaling is found for a two-particle system where the rescaling is exact. Detailed comparison between finite Hilbert space calculations and exact results for up to 5 particles show that rescaling can significantly improve the accuracy of numerical calculations in various external potentials. In addition to ground state energies, the low-lying excitation spectrum, density profile and correlation functions are studied. The results give a promising outlook for numerical simulations of trapped ultracold atoms.

Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization

In this paper, we present the implementation of efficient approximations to time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation (TDA) for hybrid density functionals. For the calculation of the TDDFT/TDA excitation energies and analytical gradients, we combine the resolution of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced "chain of spheres exchange" (COSX) algorithm for the calculation of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approximation leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 molecules with the configuration interaction singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equilibrium geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calculated excited state normal coordinate displacements is of the order of 0.01, and relative error in the calculated excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approximation are, thus, insignificant compared to the errors related to the approximate nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calculations on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ~26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ~27-29 for 30 processors). The solution of the Z-vector equations shows a speedup of ~24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ~15-25 for 30 processors), but their contribution to the total calculation time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calculation on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calculation of the TDDFT excited state properties in the frozen-core approximation. The algorithms described in this work are implemented into the ORCA electronic structure system.

Accurate thermochemistry and spectroscopy of the oxygen-protonated sulfur dioxide isomers

Despite the promising relevance of protonated sulfur dioxide in astrophysical and atmospheric fields, its thermochemical and spectroscopic characterization is very limited. High-level quantum-chemical calculations have shown that the most stable isomer is the cis oxygen-protonated sulfur dioxide, HOSO(+), while the trans form is about 2 kcal mol(-1) less stable; even less stable (by about 42 kcal mol(-1)) is the S-protonated isomer [V. Lattanzi et al., J. Chem. Phys., 2010, 133, 194305]. The enthalpy of formation for the cis- and trans-HOSO(+) is presented, based on the well tested HEAT protocol [A. Tajti et al., J. Chem. Phys., 2004, 121, 11599]. Systematically extrapolated ab initio energies, accounting for electron correlation through coupled cluster theory, including up to single, double, triple and quadruple excitations, have been corrected for core-electron correlation, anharmonic zero-point vibrational energy, diagonal Born-Oppenheimer and scalar relativistic effects. As a byproduct, proton affinity of sulfur dioxide and atomization energies have also been obtained at the same levels of theory. Vibrational and rotational spectroscopic properties have been investigated by means of composite schemes that allow us to account for truncation of basis set as well as core correlation. Where available, for both thermochemistry and spectroscopy, very good agreement with experimental data has been observed.

Interference-corrected explicitly-correlated second-order perturbation theory

In the complete-basis-set (CBS) model chemistries, an interference correction is added to the total energy. It is added because the CBS-type extrapolation to the basis-set limit of second-order Møller–Plesset (MP2) perturbation theory usually overestimates the magnitude of the basis-set truncation error at higher levels of electron-correlation treatment. In this Letter, the question is pursued whether the CBS-type interference correction could also improve the computational results when the MP2 basis set limit is approached by means of explicitly-correlated MP2 theory using Slater-type geminals.

Basis set convergence of potential energy surfaces: Ground electronic state of H2 and H

AbstractThe convergence behavior of the total energy with the increase of the basis set is analyzed in two‐electron systems, H2 and H , where the source of error is the basis set truncation. Extrapolation of total energy to the complete basis set limit is carried out with the exponential and inverse cubic power (or higher order) functional forms, as a function of the cardinal number X that describes the correlation consistent basis sets, used in this work. The ability of the different extrapolation schemes to generate accurate potential energy curves or surfaces, over a wide range of geometries, is examined, following the criteria outlined by Halkier et al. (Chem Phys Lett, 1999, 310, 385). Comparison with the most accurate potential energy curve and surface for the ground electronic state of H2 (X1Σ) and H (1A′), respectively, shows that a simple two point extrapolation with the quintuple and sextuple zeta correlation‐consistent basis sets results yields to practically “spectroscopic accuracy.” To assess the precision of the extrapolated potential energy surfaces, several properties are compared with the most accurate ones. As an example, the vibrational energies obtained with the extrapolated energies have an error of only a few tenths of a cm−1. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011