Szymon Śmiga

Comparing ab initio density-functional and wave function theories: The impact of correlation on the electronic density and the role of the correlation potential

The framework of ab initio density-functional theory (DFT) has been introduced as a way to provide a seamless connection between the Kohn-Sham (KS) formulation of DFT and wave-function based ab initio approaches [R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem. Phys. 122, 034104 (2005)]. Recently, an analysis of the impact of dynamical correlation effects on the density of the neon atom was presented [K. Jankowski, K. Nowakowski, I. Grabowski, and J. Wasilewski, J. Chem. Phys. 130, 164102 (2009)], contrasting the behaviour for a variety of standard density functionals with that of ab initio approaches based on second-order Møller-Plesset (MP2) and coupled cluster theories at the singles-doubles (CCSD) and singles-doubles perturbative triples [CCSD(T)] levels. In the present work, we consider ab initio density functionals based on second-order many-body perturbation theory and coupled cluster perturbation theory in a similar manner, for a range of small atomic and molecular systems. For comparison, we also consider results obtained from MP2, CCSD, and CCSD(T) calculations. In addition to this density based analysis, we determine the KS correlation potentials corresponding to these densities and compare them with those obtained for a range of ab initio density functionals via the optimized effective potential method. The correlation energies, densities, and potentials calculated using ab initio DFT display a similar systematic behaviour to those derived from electronic densities calculated using ab initio wave function theories. In contrast, typical explicit density functionals for the correlation energy, such as VWN5 and LYP, do not show behaviour consistent with this picture of dynamical correlation, although they may provide some degree of correction for already erroneous explicitly density-dependent exchange-only functionals. The results presented here using orbital dependent ab initio density functionals show that they provide a treatment of exchange and correlation contributions within the KS framework that is more consistent with traditional ab initio wave function based methods.

A density difference based analysis of orbital-dependent exchange-correlation functionals

We present a density difference based analysis for a range of orbital-dependent Kohn–Sham functionals. Results for atoms, some members of the neon isoelectronic series and small molecules are reported and compared with ab initio wave function calculations. Particular attention is paid to the quality of approximations to the exchange-only optimised effective potential (OEP) approach: we consider both the localised Hartree–Fock as well as the Krieger–Li–Iafrate methods. Analysis of density differences at the exchange-only level reveals the impact of the approximations on the resulting electronic densities. These differences are further quantified in terms of the ground state energies, frontier orbital energy differences and highest occupied orbital energies obtained. At the correlated level, an OEP approach based on a perturbative second-order correlation energy expression is shown to deliver results comparable with those from traditional wave function approaches, making it suitable for use as a benchmark against which to compare standard density functional approximations.

Orbital-dependent second-order scaled-opposite-spin correlation functionals in the optimized effective potential method

The performance of correlated optimized effective potential (OEP) functionals based on the spin-resolved second-order correlation energy is analysed. The relative importance of singly- and doubly- excited contributions as well as the effect of scaling the same- and opposite- spin components is investigated in detail comparing OEP results with Kohn-Sham (KS) quantities determined via an inversion procedure using accurate ab initio electronic densities. Special attention is dedicated in particular to the recently proposed scaled-opposite-spin OEP functional [I. Grabowski, E. Fabiano, and F. Della Sala, Phys. Rev. B 87, 075103 (2013)] which is the most advantageous from a computational point of view. We find that for high accuracy, a careful, system dependent, selection of the scaling coefficient is required. We analyse several size-extensive approaches for this selection. Finally, we find that a composite approach, named OEP2-SOSh, based on a post-SCF rescaling of the correlation energy can yield high accuracy for many properties, being comparable with the most accurate OEP procedures previously reported in the literature but at substantially reduced computational effort.

Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method

We introduce an orbital-optimized double-hybrid (DH) scheme using the optimized-effective-potential (OEP) method. The orbitals are optimized using a local potential corresponding to the complete exchange-correlation energy expression including the second-order Møller-Plesset correlation contribution. We have implemented a one-parameter version of this OEP-based self-consistent DH scheme using the BLYP density-functional approximation and compared it to the corresponding non-self-consistent DH scheme for calculations on a few closed-shell atoms and molecules. While the OEP-based self-consistency does not provide any improvement for the calculations of ground-state total energies and ionization potentials, it does improve the accuracy of electron affinities and restores the meaning of the LUMO orbital energy as being connected to a neutral excitation energy. Moreover, the OEP-based self-consistent DH scheme provides reasonably accurate exchange-correlation potentials and correlated densities.

Subsystem density functional theory with meta-generalized gradient approximation exchange-correlation functionals

We analyze the methodology and the performance of subsystem density functional theory (DFT) with meta-generalized gradient approximation (meta-GGA) exchange-correlation functionals for non-bonded molecular systems. Meta-GGA functionals depend on the Kohn-Sham kinetic energy density (KED), which is not known as an explicit functional of the density. Therefore, they cannot be directly applied in subsystem DFT calculations. We propose a Laplacian-level approximation to the KED which overcomes this limitation and provides a simple and accurate way to apply meta-GGA exchange-correlation functionals in subsystem DFT calculations. The so obtained density and energy errors, with respect to the corresponding supermolecular calculations, are comparable with conventional approaches, depending almost exclusively on the approximations in the non-additive kinetic embedding term. An embedding energy error decomposition explains the accuracy of our method.

Laplacian-dependent models of the kinetic energy density: Applications in subsystem density functional theory with meta-generalized gradient approximation functionals

The development of semilocal models for the kinetic energy density (KED) is an important topic in density functional theory (DFT). This is especially true for subsystem DFT, where these models are necessary to construct the required non-additive embedding contributions. In particular, these models can also be efficiently employed to replace the exact KED in meta-Generalized Gradient Approximation (meta-GGA) exchange-correlation functionals allowing to extend the subsystem DFT applicability to the meta-GGA level of theory. Here, we present a two-dimensional scan of semilocal KED models as linear functionals of the reduced gradient and of the reduced Laplacian, for atoms and weakly bound molecular systems. We find that several models can perform well but in any case the Laplacian contribution is extremely important to model the local features of the KED. Indeed a simple model constructed as the sum of Thomas-Fermi KED and 1/6 of the Laplacian of the density yields the best accuracy for atoms and weakly bound molecular systems. These KED models are tested within subsystem DFT with various meta-GGA exchange-correlation functionals for non-bonded systems, showing a good accuracy of the method.

Accurate Kohn–Sham ionization potentials from scaled-opposite-spin second-order optimized effective potential methods

One important property of Kohn–Sham (KS) density functional theory is the exact equality of the energy of the highest occupied KS orbital (HOMO) with the negative ionization potential of the system. This exact feature is out of reach for standard density‐dependent semilocal functionals. Conversely, accurate results can be obtained using orbital‐dependent functionals in the optimized effective potential (OEP) approach. In this article, we investigate the performance, in this context, of some advanced OEP methods, with special emphasis on the recently proposed scaled‐opposite‐spin OEP functional. Moreover, we analyze the impact of the so‐called HOMO condition on the final quality of the HOMO energy. Results are compared to reference data obtained at the CCSD(T) level of theory.

Jellium-with-gap model applied to semilocal kinetic functionals

Lucian A. Constantin, Eduardo Fabiano, 1 Szymon Śmiga, 4, 1 and Fabio Della Sala 1 Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy Institute for Microelectronics and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce, Italy Istituto Nanoscienze-CNR, Italy Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland (Dated: April 16, 2018)

Chapter 7 – Density-Dependent Exchange–Correlation Potentials Derived From highly Accurate Ab initio Calculations

Abstract Using electron densities and exchange–correlation potentials obtained from ab initio wave function theory (WFT) or optimized effective potential (OEP) calculations, we have derived new density-dependent exchange and exchange-correlation potentials in a context of Kohn–Sham density functional theory (KS-DFT). Using these new potentials, standard KS-DFT calculations have been performed for few small atoms. Results obtained with the new density-dependent exchange-correlation potentials are much better then standard KS-DFT results, and quantitatively and qualitatively comparable with those obtained form ab initio WFT and the correlated OEP methods, with a much lower computational cost.

Spin-Component-Scaled ΔMP2 Parametrization: Toward a Simple and Reliable Method for Ionization Energies

A practical, accurate, and cost- and implementation-free method (ΔMP2-SOS(IP)) for the calculation of vertical ionization potentials is proposed. The simple method is based on a single-step, a diagonal, frequency-independent approximation to the second-order self-energy expression combined with the spin-component-scaled technique. The search for an optimal scaling factor is performed for a set of 50 moderately sized molecules, and the quality of the method is additionally assessed for a benchmark set of 24 organic acceptor molecules. The proposed ΔMP2-SOS(IP) method provides the best results of valence ionization energies as compared to the several standard self-consistent variants of the electron propagator methods at the second and higher orders (EP2, SCS-EP2, EP3, OVGF) with almost CCSD(T) or IP-EOM-CCSD accuracy and the cost of only a single opposite-spin ΔMP2-type calculation ( O( N3)). For core ionization energies, our new methods outperform the standard ΔMP2 results due to a better balanced treatment of the correlation and relaxation term in the second-order self-energy.