The direct approach to gravitation and electrostatics method for periodic systems

Losilla, Sergio A.; Sundholm, Dage; Jusélius, Jonas

doi:10.1063/1.3291027

Cited by 26 publications

(28 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(5), we employed the interpolating scaling function method which has been originally used to compute Hartree potentials. [45][46][47][48][49][50] First, Eq. (5) can be rewritten as…”

Section: Cis Based On Lagrange-sinc Functionsmentioning

confidence: 99%

Configuration interaction singles based on the real-space numerical grid method: Kohn–Sham versus Hartree–Fock orbitals

Kim

Hong

Choi

et al. 2015

Phys. Chem. Chem. Phys.

View full text Add to dashboard Cite

We developed a program code of configuration interaction singles (CIS) based on a numerical grid method. We used Kohn-Sham (KS) as well as Hartree-Fock (HF) orbitals as a reference configuration and Lagrange-sinc functions as a basis set. Our calculations show that KS-CIS is more cost-effective and more accurate than HF-CIS. The former is due to the fact that the non-local HF exchange potential greatly reduces the sparsity of the Hamiltonian matrix in grid-based methods. The latter is because the energy gaps between KS occupied and virtual orbitals are already closer to vertical excitation energies and thus KS-CIS needs small corrections, whereas HF results in much larger energy gaps and more diffuse virtual orbitals. KS-CIS using the Lagrange-sinc basis set also shows a better or a similar accuracy to smaller orbital space compared to the standard HF-CIS using Gaussian basis sets. In particular, KS orbitals from an exact exchange potential by the Krieger-Li-Iafrate approximation lead to more accurate excitation energies than those from conventional (semi-) local exchange-correlation potentials.

show abstract

“…(5), we employed the interpolating scaling function method which has been originally used to compute Hartree potentials. [45][46][47][48][49][50] First, Eq. (5) can be rewritten as…”

Section: Cis Based On Lagrange-sinc Functionsmentioning

confidence: 99%

Configuration interaction singles based on the real-space numerical grid method: Kohn–Sham versus Hartree–Fock orbitals

Kim

Hong

Choi

et al. 2015

Phys. Chem. Chem. Phys.

View full text Add to dashboard Cite

show abstract

“…16 The radial part of one-center potentials is calculated by using inward and outward integrations of the radial densities, whereas the cube part is obtained by a direct 3D numerical integration. [13][14][15] For the integration of the Poisson kernel, the Coulomb operator is rewritten using the integral expression and discretized as 1…”

Section: Integration Of the Poisson Kernelmentioning

confidence: 99%

“…(14). [13][14][15][16] The expansion coefficients f ∆ i j k of the cube are obtained by projecting out the steep bubble contribution from f (r) as described above for ρ(r) in the case of the Poisson kernel. The methods of the projection are described in detail in Ref.…”

Section: Integration Of the Helmholtz Kernelmentioning

confidence: 99%

“…The linear region [0, t l ] is integrated using ordinary Gaussian quadrature, the [t l , t f ] region is integrated using Gaussian quadrature in logarithmic coordinates, and the integration in the interval of [t f , ∞] is calculated analytically leading to an error of the order of t −4 f . 15 For the Helmholtz kernel, the shape of the t integrand, i.e., the F k (t) function in Eq. (23) can be obtained from the normalization condition of ground-state hydrogenlike wave functions using different values for k corresponding to the orbital energy, which is analogous to the approach we used when investigating the t dependence of the Coulomb kernel, 15…”

Section: Integration In the T-spacementioning

confidence: 99%

“…In this work, we have implemented an approach to optimize numerical one-particle wave functions of bound electronic states by integrating the Helmholtz kernel using a similar approach as we previously employed for calculating electrostatic potentials using the Poisson kernel, i.e., integrating the Coulomb integral. [13][14][15][16][17] The singular attraction potential between the electrons and the nuclei of molecular systems leads to very steep cusps of the orbitals and potentials in the vicinity of the nuclei. Since the steep cusps are difficult to expand using numerical methods, we have devised the use of a double numerical basis, where the steep parts of the functions are expanded in one-center functions at each nucleus, whereas the remainder is described using a three-dimensional (3D) equidistant grid.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimization of numerical orbitals using the Helmholtz kernel

Solala

Losilla

Sundholm

et al. 2017

The Journal of Chemical Physics

View full text Add to dashboard Cite

We present an integration scheme for optimizing the orbitals in numerical electronic structure calculations on general molecules. The orbital optimization is performed by integrating the Helmholtz kernel in the double bubble and cube basis, where bubbles represent the steep part of the functions in the vicinity of the nuclei, whereas the remaining cube part is expanded on an equidistant three-dimensional grid. The bubbles' part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kinds. The angular part of the bubble functions can be integrated analytically, whereas the radial part is integrated numerically. The cube part is integrated using a similar method as we previously implemented for numerically integrating two-electron potentials. The behavior of the integrand of the auxiliary dimension introduced by the integral transformation of the Helmholtz kernel has also been investigated. The correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations on H, HO, and CO. The obtained energies are compared with reference values in the literature showing that an accuracy of 10 to 10 E can be obtained with our approach.

show abstract

Performance of heterogeneous computing with graphics processing unit and many integrated core for hartree potential calculations on a numerical grid

et al. 2016

View full text Add to dashboard Cite

We investigated the performance of heterogeneous computing with graphics processing units (GPUs) and many integrated core (MIC) with 20 CPU cores (20×CPU). As a practical example toward large scale electronic structure calculations using grid-based methods, we evaluated the Hartree potentials of silver nanoparticles with various sizes (3.1, 3.7, 4.9, 6.1, and 6.9 nm) via a direct integral method supported by the sinc basis set. The so-called work stealing scheduler was used for efficient heterogeneous computing via the balanced dynamic distribution of workloads between all processors on a given architecture without any prior information on their individual performances. 20×CPU + 1GPU was up to ∼1.5 and ∼3.1 times faster than 1GPU and 20×CPU, respectively. 20×CPU + 2GPU was ∼4.3 times faster than 20×CPU. The performance enhancement by CPU + MIC was considerably lower than expected because of the large initialization overhead of MIC, although its theoretical performance is similar with that of CPU + GPU. © 2016 Wiley Periodicals, Inc.

show abstract

The direct approach to gravitation and electrostatics method for periodic systems

Cited by 26 publications

References 22 publications

Configuration interaction singles based on the real-space numerical grid method: Kohn–Sham versus Hartree–Fock orbitals

Configuration interaction singles based on the real-space numerical grid method: Kohn–Sham versus Hartree–Fock orbitals

Optimization of numerical orbitals using the Helmholtz kernel

Performance of heterogeneous computing with graphics processing unit and many integrated core for hartree potential calculations on a numerical grid

Contact Info

Product

Resources

About