Mixed-basis Approach Research Articles

Density functional calculations of atomic structure, charging effect, and static dielectric constant of two-dimensional systems based on B-splines

We have implemented a total-energy minimization scheme to allow for relaxation of atomic positions in density functional calculations for two-dimensional (2D) materials using a mixed basis set. The basis functions consist of products of 2D plane waves in the plane of the material and localized B-splines along the perpendicular direction. By using the mixed basis approach (MBA), we have studied the atomic relaxation and charge polarization of 2D systems under an applied electric field. In contrast to the conventional super-cell approach (SCA) which adopts repeated slabs sandwiched between vacuum regions, MBA makes no requirement of compensating background charge for treating electrically charged 2D systems due to carrier injection. With MBA we can avoid using a periodic sawtooth potential for systems under the applied field as commonly adopted in SCA. Furthermore, we introduced a simple method to determine the out-of-plane dielectric constants of 2D materials without the ambiguity of defining their effective thickness. This is achieved by calculating the linear response in charge polarization to the applied field. Selected 2D systems, including graphene and transition-metal dichalcogenides are tested. Our MBA results are consistent with previous SCA calculations when both approaches are equally applicable. However, for charged systems with high carrier density, we find significant deviation from SCA results obtained by imposing artificial charge neutrality condition.

Comprehensive Basis-Set Testing of Extended Symmetry-Adapted Perturbation Theory and Assessment of Mixed-Basis Combinations to Reduce Cost.

Hybrid or "extended" symmetry-adapted perturbation theory (XSAPT) replaces traditional SAPT's treatment of dispersion with better performing alternatives while at the same time extending two-body (dimer) SAPT to a many-body treatment of polarization using a self-consistent charge embedding procedure. The present work presents a systematic study of how XSAPT interaction energies and energy components converge with respect to the choice of Gaussian basis set. Errors can be reduced in a systematic way using correlation-consistent basis sets, with aug-cc-pVTZ results converged within <0.1 kcal/mol. Similar (if slightly less systematic) behavior is obtained using Karlsruhe basis sets at much lower cost, and we introduce new versions with limited augmentation that are even more efficient. Pople-style basis sets, which are more efficient still, often afford good results if a large number of polarization functions are included. The dispersion models used in XSAPT afford much faster basis-set convergence as compared to the perturbative description of dispersion in conventional SAPT, meaning that "compromise" basis sets (such as jun-cc-pVDZ) are no longer required and benchmark-quality results can be obtained using triple-ζ basis sets. The use of diffuse functions proves to be essential, especially for the description of hydrogen bonds. The "δ(Hartree-Fock)" correction for high-order induction can be performed in double-ζ basis sets without significant loss of accuracy, leading to a mixed-basis approach that offers 4× speedup over the existing (cubic scaling) XSAPT approach.

Gaussian-Type Orbital Calculations for High Harmonic Generation in Vibrating Molecules: Benchmarks for H2.

The response of the hydrogen molecular ion, H2+, to few-cycle laser pulses of different intensities is simulated. To treat the coupled electron-nuclear motion, we use adiabatic potentials computed with Gaussian-type basis sets together with a heuristic ionization model for the electron and a grid representation for the nuclei. Using this mixed-basis approach, the time-dependent Schrödinger equation is solved, either within the Born-Oppenheimer approximation or with nonadiabatic couplings included. The dipole response spectra are compared to all-grid-based solutions for the three-body problem, which we take as a reference to benchmark the Gaussian-type basis set approaches. Also, calculations employing the fixed-nuclei approximation are performed, to quantify effects due to nuclear motion. For low intensities and small ionization probabilities, we get excellent agreement of the dynamics using Gaussian-type basis sets with the all-grid solutions. Our investigations suggest that high harmonic generation (HHG) and high-frequency response, in general, can be reliably modeled using Gaussian-type basis sets for the electrons for not too high harmonics. Further, nuclear motion destroys electronic coherences in the response spectra even on the time scale of about 30 fs and affects HHG intensities, which reflect the electron dynamics occurring on the attosecond time scale. For the present system, non-Born-Oppenheimer effects are small. The Gaussian-based, nonadiabatically coupled, time-dependent multisurface approach to treat quantum electron-nuclear motion beyond the non-Born-Oppenheimer approximation can be easily extended to approximate wavefunction methods, such as time-dependent configuration interaction singles (TD-CIS), for systems where no benchmarks are available.

Chirality of small niobium clusters

We report on time-dependent density functional theory calculations to study the chiroptical properties of small $$\hbox {Nb}_{{9}}$$ clusters and their corresponding van der Waals $$\hbox {Nb}_{{9}}\hbox {Ar}_{n}$$ complexes. The three most stable 9-atom structures considered in the literature are analyzed, and we obtain their lowest-energy atomic configurations, UV–Vis response, and for the first time, the electronic circular dichroism (ECD) spectra of the systems. Excitation energies, as well as oscillator and rotational strengths are calculated using the M06 exchange-correlation functional together with a mixed basis approach where all electron DGDZVP and Def2SV basis are employed for Nb and Ar atoms, respectively. In all cases, the UV–Vis spectra look very similar and are characterized by the formation of a main absorption maxima located at $$\sim 240\hbox { nm}$$ ; however, the distribution of simulated ECD signals strongly depends on the geometrical structure of the clusters. Our calculated ECD bands show precise features that cover the experimentally accessible 200–300 nm range and extend near the visible part of the spectra, a fact that allows us to distinguish between different niobium cluster geometries. However, weighted averages of calculated ECD spectra for different $$\hbox {Nb}_{{9}}$$ clusters reveal that isomerization effects might lead to a notable increase/reduction in the intensity of CD lines since the simulated bands could add or cancel each other.

Creation of Mo/Tc@C60 and Au@C60 and molecular-dynamics simulations.

The formation of middle- and/or high-weight atom (Mo, Au)-incorporated fullerenes was investigated using radionuclides produced by nuclear reactions. From the trace radioactivities of 99Mo/99mTc or 194Au after high-performance liquid chromatography, it was found that the formation of endohedral and/or heterofullerene fullerenes in 99Mo/99mTc and 194Au atoms could occur by a recoil process following the nuclear reactions. Furthermore, the 99mTc (and 194Au) atoms recoiled against β-decay remained present inside these cages. To confirm the produced materials experimentally, ab initio molecular dynamics (MD) simulations based on an all-electron mixed-basis approach were performed. The possibility of the formation of endohedral fullerenes containing Mo/Tc and Au atoms is verified; here, the formation of heterofullerenes is excluded by MD simulations. These findings suggest that radionuclides stably encapsulated by fullerenes could potentially play a valuable role in diagnostic nuclear medicine.

Implementation of hyperfine coupling in all-electron mixed basis approach

In order to accurately evaluate hyperfine interaction (HFI) (both the isotropic, Fermi contact term and the anisotropic, dipole interaction term) of spin-polarized atoms and molecules, we newly implement the cubic harmonics routines for calculating HFI in the all-electron mixed basis code, TOMBO, in which both plane waves (PWs) and atomic orbitals (AOs) are used as basis functions, and AOs are restricted inside the non-overlapping atomic spheres and composed of the numerical radial function on a logarithmic radial mesh and analytic cubic harmonics. The method has a distinct merit of describing the cusp behavior near the nucleus position, which is essential for the Fermi contact term, and of significantly reducing the required cutoff energy for PWs compared to the usual PW expansion approach accompanied with the pseudopotential or PAW method. We also considered the contributions from surrounding atoms in the dipole interaction. Calculated results show excellent agreement with the experimental data, indicating the excellent performance of the code.

Spin‐Orbit Coupling in All‐Electron Mixed Basis Approach

AbstractIn the evaluation of the spin‐orbit coupling (SOC), the use of the L · S formula is invalid in the interatomic region where the effective potential is not spherically symmetric. This problem occurs in the LCAO, LMTO, and APW methods, while the plane‐wave pseudopotential and PAW methods cannot treat the spin‐orbit splitting (SOS) of core orbitals. To avoid these problems, the all‐electron mixed basis approach is adopted, which uses both plane waves (PWs) and atomic orbitals (AOs) as a basis set. The general form S · ∇V × p can be used for PWs, while the standard form L · S can be used for AOs, which are well localized inside the non‐overlapping atomic sphere in the spherical potential region and composed of the numerical radial function on a logarithmic radial mesh and analytic cubic harmonics. The explicit formula of the AO–AO, PW–AO, and PW–PW matrix elements of the SOC for spin‐polarized systems is presented. In particular, the AO–AO matrices are explicitly derived for p, d, and f orbitals. The method is applied to the SOS of core and valence levels in X‐ray photoelectron spectra. The results are in excellent agreement with the available experimental data, which suggests the validity of the present method.

Application of van der Waals density functionals to two dimensional systems based on a mixed basis approach

A van der Waals (vdW) density functional was implemented with the charge density obtained by the mixed basis approach previously developed for studying two dimensional systems, in which the vdW interaction plays an important role. The basis functions here are taken to be the localized B-splines for the finite non-periodic dimension and plane waves for the two periodic directions. This approach will reduce the size of the basis set, especially for large systems, and therefore is computationally efficient for the diagonalization of the Kohn–Sham Hamiltonian. The nonlocal vdW correlation was treated appropriately according to the layered geometry structure. We applied the present algorithm to calculate the binding energy for graphene and hexagonal boron nitride and the results are consistent with data reported earlier. We also found that, due to the relatively weak vdW interaction, the charge density obtained self-consistently for the whole bilayer system is not significantly different from the simple addition of those for the two individual monolayer systems, except when the interlayer separation is close enough that the strong electron-repulsion dominates. This finding suggests an efficient way to calculate the vdW interaction for large complex systems involving Moiré pattern configurations.

Optical properties of six isomers of three dimensionally delocalized π-conjugated carbon nanocage

First-principles GW+Bethe-Salpeter method employing all-electron mixed basis approach is applied to hydrocarbon molecules consisting of 78–198 atoms and its theoretical accuracy and performance are evaluated. Based on the confirmed accuracy/reliability of our method, we simulated the UV–vis absorption spectra of previously reported six possible isomers [E. Kayahara et al., Nat. Chem. 4, 2694 (2013)]. We also attempted to identify the most stable isomers of recently synthesized ball-shaped carbon nanocages by taking into account available experimental spectra. The best agreement with the experiment is found for the most unstable isomer, labelled as T. Our simulation strongly suggests that the external experimental conditions such as solution and finite temperature affect stability.

Ab initio molecular dynamics simulation study of successive hydrogenation reactions of carbon monoxide producing methanol.

Doing ab initio molecular dynamics simulations, we demonstrate a possibility of hydrogenation of carbon monoxide producing methanol step by step. At first, the hydrogen atom reacts with the carbon monoxide molecule at the excited state forming the formyl radical. Formaldehyde was formed after adding one more hydrogen atom to the system. Finally, absorption of two hydrogen atoms to formaldehyde produces methanol molecule. This study is performed by using the all-electron mixed basis approach based on the time dependent density functional theory within the adiabatic local density approximation for an electronic ground-state configuration and the one-shot GW approximation for an electronic excited state configuration.

A mixed basis density functional approach for one-dimensional systems with B-splines

A mixed basis approach based on density functional theory is extended to one-dimensional (1D) systems. The basis functions here are taken to be the localized B-splines for the two finite non-periodic dimensions and the plane waves for the third periodic direction. This approach will significantly reduce the number of the basis and therefore is computationally efficient for the diagonalization of the Kohn–Sham Hamiltonian. For 1D systems, B-spline polynomials are particularly useful and efficient in two-dimensional spatial integrations involved in the calculations because of their absolute localization. Moreover, B-splines are not associated with atomic positions when the geometry structure is optimized, making the geometry optimization easy to implement. With such a basis set we can directly calculate the total energy of the isolated system instead of using the conventional supercell model with artificial vacuum regions among the replicas along the two non-periodic directions. The spurious Coulomb interaction between the charged defect and its repeated images by the supercell approach for charged systems can also be avoided. A rigorous formalism for the long-range Coulomb potential of both neutral and charged 1D systems under the mixed basis scheme will be derived. To test the present method, we apply it to study the infinite carbon-dimer chain, graphene nanoribbon, carbon nanotube and positively-charged carbon-dimer chain. The resulting electronic structures are presented and discussed in detail.

A mixed basis density functional approach for low dimensional systems with B-splines

A mixed basis approach based on density functional theory is employed for low dimensional systems. The basis functions are taken to be plane waves for the periodic direction multiplied by B-spline polynomials in the non-periodic direction. B-splines have the following advantages: (1) the associated matrix elements are sparse, (2) B-splines possess a superior treatment of derivatives, (3) B-splines are not associated with atomic positions when the geometry structure is optimized, making the geometry optimization easy to implement. With this mixed basis set we can directly calculate the total energy of the system instead of using the conventional supercell model with a slab sandwiched between vacuum regions. A generalized Lanczos–Krylov iterative method is implemented for the diagonalization of the Hamiltonian matrix. To demonstrate the present approach, we apply it to study the C(001)-(2×1) surface with the norm-conserving pseudopotential, the n-type δ-doped graphene, and graphene nanoribbon with Vanderbilt’s ultra-soft pseudopotentials. All the resulting electronic structures were found to be in good agreement with those obtained by the VASP code, but with a reduced number of basis.

TOMBO: All-electron mixed-basis approach to condensed matter physics

TOMBO is a computer code for calculating the electronic structure of systems that consist both of core and valence electrons and nuclei, based on density-functional theory. It is based on an all-electron mixed-basis approach, in which the Kohn–Sham (KS) wave function is expressed by a linear combination of plane-waves and atomic-orbitals. This approach can describe both spatially localized and extended orbitals, which enables us to perform all-electron calculations with high accuracy from isolated clusters to periodic crystals. The present paper describes a theory of the all-electron mixed-basis approach, as well as input variables and benchmark tests in TOMBO. The algorithm for accelerating the computational time that is needed to solve the KS equation is also presented.

A mixed basis approach in the SGP-limit

A perturbation method for computing quick estimates of the echo decay in pulsed spin echo gradient NMR diffusion experiments in the short gradient pulse limit is presented. The perturbation basis involves (relatively few) dipole distributions on the boundaries generating a small perturbation matrix in O( s 2) time, where s denotes the number of boundary elements. Several approximate eigenvalues and eigenfunctions to the diffusion operator are retrieved. The method is applied to 1D and 2D systems with Neumann boundary conditions.

Evaluating the electronic energy eigenvalues for A2B6 crystals obtained with the GGGA+U functional

AbstractElectronic energy bands of ZnS, ZnSe, and ZnTe crystals evaluated within the mixed basis approach with using the core Bloch states and plane waves. The exchange‐correlation potential has been derived from the new exchange‐correlation‐energy functional including the density gradient corrections and the intra‐atomic Coulomb term U. The calculated energy band structure parameters of the crystals show an improvement compared with the LDA results. The semicore Zn 3d state energies are closer to experimental data than those obtained within the LDA approach. (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

All-electron first-principlesGW+Bethe-Salpeter calculation for optical absorption spectra of sodium clusters

The optical absorption spectra of sodium clusters (Na{sub 2n}, n{<=} 4) are calculated by using an all-electron first-principles GW+Bethe-Salpeter method with the mixed-basis approach within the Tamm-Dancoff approximation. In these small systems, the excitonic effect strongly affects the optical properties due to the confinement of exciton in the small system size. The present state-of-the-art method treats the electron-hole two-particle Green's function by incorporating the ladder diagrams up to the infinite order and therefore takes into account the excitonic effect in a good approximation. We check the accuracy of the present method by comparing the resulting spectra with experiments. In addition, the effect of delocalization in particular in the lowest unoccupied molecular orbital in the GW quasiparticle wave function is also discussed by rediagonalizing the Dyson equation.

All-Electron &lt;I&gt;GW&lt;/I&gt; Calculations of Silicon, Diamond, and Silicon Carbide

All-electron GW calculations of Si, diamond, and SiC were carried out to obtain the quasiparticle energy spectra, for the first time, including both valence and core levels, corresponding to the ultraviolet and X-ray photoemission spectra (UPS and XPS). We used an all-electron mixed basis approach, in which wave functions are expanded with both plane waves and numerical atomic orbitals. In particular for the core states, our results obtained using the perturbative (non-self-consistent) GW approach are in good agreement with available experimental data. The quasiparticle energies of core levels so obtained are also in good agreement with experiments when we do not take into account the renormalization factor z. Moreover, we refer to the bandwidth of diamond and compare with experiments.

Quasiparticle energy spectra of alkali-metal clusters: All-electron first-principles calculations

A good approximation to the one-electron self-energy operator in the calculation of quasiparticle energy spectra including the first ionization potential (IP) and electron affinity (EA) is to expand it as a simple product of a one-particle Green's function G and a dynamically screened Coulomb interaction W, namely, GW approximation. We developed a spin-polarized version of the all-electron GW approach and applied it to the first-principles calculation of quasiparticle energy spectra of alkali-metal clusters (Na(n) and K(n), n=1-8). Our all-electron mixed basis approach, in which wave functions are expressed as a linear combination of numerical atomic orbitals and plane waves, enables us to compare the absolute values of the singly (or highest) occupied molecular orbital and the lowest unoccupied molecular orbital levels with available experimental IPs and EAs. The agreements with the corresponding experimental values are fairly good. Comparing with the non-spin-polarized results of Na(2n) and K(2n) (n=1-4), we discuss the effect of spin polarization as well as the cluster size dependence of IPs or EAs.

First-principlesGWcalculations of GaAs clusters and crystal using an all-electron mixed basis approach

All-electron $GW$ calculations beginning with the local density approximation of the density functional theory are carried out for both small gallium arsenide clusters and gallium arsenide crystal in a consistent way by means of the all-electron mixed basis approach, in which both plane waves and atomic orbitals are used as a basis set. This approach has the merit of expressing both core electron states and empty free-electron-like states accurately with a rather small number of basis functions. The present all-electron $GW$ method enables us to treat both clusters and crystal without difficulty. First, we determined the most stable structures of gallium arsenide clusters, ${\mathrm{Ga}}_{n}{\mathrm{As}}_{n}$ with $n=2--4$. For these clusters, since the structural change in the ionization process is not so large compared to silicon or germanium clusters, there is no significant difference in the vertical and adiabatic electron affinities. For both clusters and crystal, our one-shot $GW$ results for quasiparticle energy spectra agree fairly well with available experimental photoemission and inverse photoemission data. The present results include the semirelativistic effect.

Electronic energy band parameters of CsCl evaluated on core Bloch states and plane waves

Electronic energy bands of CsCl crystal have been calculated within the mixed basis approach with using the core Bloch states and plane waves. The calculated energy parameters of the crystal are in the satisfactory agreement with the experimental data obtained from the analysis of the core–valence luminescence spectra. The obtained results form a base for calculation of CVL spectra parameters.