An Overview of Self-Consistent Field Calculations Within Finite Basis Sets

Lehtola, Susi; Blockhuys, Frank; Alsenoy, C. Van

doi:10.3390/molecules25051218

Cited by 58 publications

(57 citation statements)

References 105 publications

(135 reference statements)

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“…This is not a severe restriction in practice since numerical instabilities usually do not occur when all eigenvalues are larger than ϵ D = 10 –6 –10 –7 . 198 − 200 …”

Section: Theorymentioning

confidence: 99%

Low-Order Scaling G₀W₀ by Pair Atomic Density Fitting

Förster

Visscher

2020

J. Chem. Theory Comput.

172

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We derive a low-scaling G 0 W 0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green’s function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G 0 W 0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

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“…This is not a severe restriction in practice since numerical instabilities usually do not occur when all eigenvalues are larger than ϵ D = 10 –6 –10 –7 . 198 − 200 …”

Section: Theorymentioning

confidence: 99%

Low-Order Scaling G₀W₀ by Pair Atomic Density Fitting

Förster

Visscher

2020

J. Chem. Theory Comput.

172

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“… 12 In Libxc , the derivatives of the functional are evaluated analytically using the Maple symbolic algebra program, as is the case for all other functionals in Libxc as well. Combined with a basis set, these derivatives can be used to minimize the total energy variationally with respect to the orbital coefficients within a self-consistent field approach; we refer to ref ( 13 ) for discussion.…”

Section: Computational Detailsmentioning

confidence: 99%

“…This divergence causes convergence problems. Assuming an orthonormal basis set {χ μ }, the potentials v n σ and v τ σ contribute to the Kohn–Sham–Fock matrix as 13 The tentative physical interpretation of the divergent negative potentials is that displacing electron density toward r → ∞ would lead to a decrease in the energy. Now, if a Gaussian-type or Slater-type orbital basis set is employed, χ μ and its gradient will decay asymptotically as exp(−α μ r 2 ) or exp(−ζ μ r ), respectively, where α μ and ζ μ are the Gaussian- and Slater-type exponents, with analogous expressions for ∇χ ν .…”

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confidence: 99%

Meta-Local Density Functionals: A New Rung on Jacob’s Ladder

Lehtola

Marques

2021

J. Chem. Theory Comput.

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The homogeneous electron gas (HEG) is a key ingredient in the construction of most exchange-correlation functionals of density-functional theory. Often, the energy of the HEG is parameterized as a function of its spin density n σ , leading to the local density approximation (LDA) for inhomogeneous systems. However, the connection between the electron density and kinetic energy density of the HEG can be used to generalize the LDA by evaluating it on a geometric average n σ avg ( r ) = n σ 1– x ( r ) ñ σ x ( r ) of the local spin density n σ ( r ) and the spin density ñ σ ( r ) of a HEG that has the local kinetic energy density τ σ ( r ) of the inhomogeneous system. This leads to a new family of functionals that we term meta-local density approximations (meta-LDAs), which are still exact for the HEG, which are derived only from properties of the HEG and which form a new rung of Jacob’s ladder of density functionals [ AIP Conf. Proc. 2001 577 1 ]. The first functional of this ladder, the local τ approximation (LTA) of Ernzerhof and Scuseria [ J. Chem. Phys. 1999 111 911 ] that corresponds to x = 1 is unfortunately not stable enough to be used in self-consistent field calculations because it leads to divergent potentials, as we show in this work. However, a geometric averaging of the LDA and LTA densities with smaller values of x not only leads to numerical stability of the resulting functional but also yields more accurate exchange energies in atomic calculations than the LDA, the LTA, or the tLDA functional ( x = 1/4) of Eich and Hellgren [ 25494732 J. Chem. Phys. 2014 141 224107 ]. We choose x = 0.50, as it gives the best total energy in self-consistent exchange-only calculations for the argon atom. Atomization energy benchmarks confirm that the choice x = 0.50 also yields improved energetics in combination with correlation functionals in molecules, almost eliminating the well-known overbinding of the LDA and reducing its error by two thirds.

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“…The DO with exponential transformation is more general than SCF, since it can be applied to both unitary and non-unitary invariant functionals, such as SIC functionals 46 . Furthermore, the anti-Hermitian matrices form a linear space making it possible to employ quasi-Newton unconstrained optimization strategies 57 , with the advan-tage that gradient-based optimization guarantees more rigorous convergence than SCF 39,58 . Quasi-Newton methods can locate nth-order saddle points of a multidimensional surface if the initial guess is sufficiently good and the formula chosen to compute the search direction is able to build an approximation to the Hessian with the appropriate number of negative eigenvalues.…”

Section: Theorymentioning

confidence: 99%