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Selenius E, Sigurdarson AE, Schmerwitz YLA, Levi G. Orbital-Optimized Versus Time-Dependent Density Functional Calculations of Intramolecular Charge Transfer Excited States. J Chem Theory Comput 2024; 20:3809-3822. [PMID: 38695313 DOI: 10.1021/acs.jctc.3c01319] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/15/2024]
Abstract
The performance of time-independent, orbital-optimized calculations of excited states is assessed with respect to charge transfer excitations in organic molecules in comparison to the linear-response time-dependent density functional theory (TD-DFT) approach. A direct optimization method to converge on saddle points of the electronic energy surface is used to carry out calculations with the local density approximation (LDA) and the generalized gradient approximation (GGA) functionals PBE and BLYP for a set of 27 excitations in 15 molecules. The time-independent approach is fully variational and provides a relaxed excited state electron density from which the extent of charge transfer is quantified. The TD-DFT calculations are generally found to provide larger charge transfer distances compared to the orbital-optimized calculations, even when including orbital relaxation effects with the Z-vector method. While the error on the excitation energy relative to theoretical best estimates is found to increase with the extent of charge transfer up to ca. -2 eV for TD-DFT, no correlation is observed for the orbital-optimized approach. The orbital-optimized calculations with the LDA and the GGA functionals provide a mean absolute error of ∼0.7 eV, outperforming TD-DFT with both local and global hybrid functionals for excitations with a long-range charge transfer character. Orbital-optimized calculations with the global hybrid functional B3LYP and the range-separated hybrid functional CAM-B3LYP on a selection of states with short- and long-range charge transfer indicate that inclusion of exact exchange has a small effect on the charge transfer distance, while it significantly improves the excitation energy, with the best-performing functional CAM-B3LYP providing an absolute error typically around 0.15 eV.
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Affiliation(s)
- Elli Selenius
- Science Institute of the University of Iceland, Reykjavík 107, Iceland
| | | | | | - Gianluca Levi
- Science Institute of the University of Iceland, Reykjavík 107, Iceland
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2
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Sethio D, Azzopardi E, Fdez. Galván I, Lindh R. A Story of Three Levels of Sophistication in SCF/KS-DFT Orbital Optimization Procedures. J Phys Chem A 2024; 128:2472-2486. [PMID: 38483190 PMCID: PMC10983011 DOI: 10.1021/acs.jpca.3c07647] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2023] [Revised: 02/23/2024] [Accepted: 02/23/2024] [Indexed: 04/04/2024]
Abstract
In this work, three versions of self-consistent field/Kohn-Sham density functional theory (SCF/KS-DFT) orbital optimization are described and benchmarked. The methods are a modified version of the geometry version of the direct inversion in the iterative subspace approach (which we call r-GDIIS), the modified restricted step rational function optimization method (RS-RFO), and the novel subspace gradient-enhanced Kriging method combined with restricted variance optimization (S-GEK/RVO). The modifications introduced are aimed at improving the robustness and computational scaling of the procedures. In particular, the subspace approach in S-GEK/RVO allows the application to SCF/KS-DFT optimization of a machine learning technique that has proven to be successful in geometry optimizations. The performance of the three methods is benchmarked for a large number of small- to medium-sized organic molecules, at equilibrium structures and close to a transition state, and a second set of molecules containing closed- and open-shell transition metals. The results indicate the importance of the resetting technique in boosting the performance of the r-GDIIS procedure. Moreover, it is demonstrated that already at the inception of the subspace version of GEK to optimize SCF wave functions, it displays superior and robust convergence properties as compared to those of the standard state-of-the-art SCF/KS-DFT optimization methods.
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Affiliation(s)
- Daniel Sethio
- Department
of Chemistry—BMC, Uppsala University, P.O. Box 576, SE-75123 Uppsala, Sweden
- Department
of Chemistry—Ångström, Uppsala University, P.O. Box 538, SE-75121 Uppsala, Sweden
| | - Emily Azzopardi
- Department
of Chemistry—BMC, Uppsala University, P.O. Box 576, SE-75123 Uppsala, Sweden
| | - Ignacio Fdez. Galván
- Department
of Chemistry—BMC, Uppsala University, P.O. Box 576, SE-75123 Uppsala, Sweden
| | - Roland Lindh
- Department
of Chemistry—BMC, Uppsala University, P.O. Box 576, SE-75123 Uppsala, Sweden
- Uppsala
Center for Computational Chemistry (UC3), Uppsala University, P.O. Box 576, SE-751 23 Uppsala, Sweden
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3
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Mortensen JJ, Larsen AH, Kuisma M, Ivanov AV, Taghizadeh A, Peterson A, Haldar A, Dohn AO, Schäfer C, Jónsson EÖ, Hermes ED, Nilsson FA, Kastlunger G, Levi G, Jónsson H, Häkkinen H, Fojt J, Kangsabanik J, Sødequist J, Lehtomäki J, Heske J, Enkovaara J, Winther KT, Dulak M, Melander MM, Ovesen M, Louhivuori M, Walter M, Gjerding M, Lopez-Acevedo O, Erhart P, Warmbier R, Würdemann R, Kaappa S, Latini S, Boland TM, Bligaard T, Skovhus T, Susi T, Maxson T, Rossi T, Chen X, Schmerwitz YLA, Schiøtz J, Olsen T, Jacobsen KW, Thygesen KS. GPAW: An open Python package for electronic structure calculations. J Chem Phys 2024; 160:092503. [PMID: 38450733 DOI: 10.1063/5.0182685] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2023] [Accepted: 01/15/2024] [Indexed: 03/08/2024] Open
Abstract
We review the GPAW open-source Python package for electronic structure calculations. GPAW is based on the projector-augmented wave method and can solve the self-consistent density functional theory (DFT) equations using three different wave-function representations, namely real-space grids, plane waves, and numerical atomic orbitals. The three representations are complementary and mutually independent and can be connected by transformations via the real-space grid. This multi-basis feature renders GPAW highly versatile and unique among similar codes. By virtue of its modular structure, the GPAW code constitutes an ideal platform for the implementation of new features and methodologies. Moreover, it is well integrated with the Atomic Simulation Environment (ASE), providing a flexible and dynamic user interface. In addition to ground-state DFT calculations, GPAW supports many-body GW band structures, optical excitations from the Bethe-Salpeter Equation, variational calculations of excited states in molecules and solids via direct optimization, and real-time propagation of the Kohn-Sham equations within time-dependent DFT. A range of more advanced methods to describe magnetic excitations and non-collinear magnetism in solids are also now available. In addition, GPAW can calculate non-linear optical tensors of solids, charged crystal point defects, and much more. Recently, support for graphics processing unit (GPU) acceleration has been achieved with minor modifications to the GPAW code thanks to the CuPy library. We end the review with an outlook, describing some future plans for GPAW.
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Affiliation(s)
- Jens Jørgen Mortensen
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Ask Hjorth Larsen
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Mikael Kuisma
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Aleksei V Ivanov
- Riverlane Ltd., St Andrews House, 59 St Andrews Street, Cambridge CB2 3BZ, United Kingdom
| | - Alireza Taghizadeh
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Andrew Peterson
- School of Engineering, Brown University, Providence, Rhode Island 02912, USA
| | - Anubhab Haldar
- Department of Electrical and Computer Engineering, Boston University, Boston, Massachusetts 02215, USA
| | - Asmus Ougaard Dohn
- Department of Physics, Technical University of Denmark, 2800 Lyngby, Denmark and Science Institute and Faculty of Physical Sciences, VR-III, University of Iceland, Reykjavík 107, Iceland
| | - Christian Schäfer
- Department of Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
| | - Elvar Örn Jónsson
- Science Institute and Faculty of Physical Sciences, University of Iceland, VR-III, 107 Reykjavík, Iceland
| | - Eric D Hermes
- Quantum-Si, 29 Business Park Drive, Branford, Connecticut 06405, USA
| | | | - Georg Kastlunger
- CatTheory, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Gianluca Levi
- Science Institute and Faculty of Physical Sciences, University of Iceland, VR-III, 107 Reykjavík, Iceland
| | - Hannes Jónsson
- Science Institute and Faculty of Physical Sciences, University of Iceland, VR-III, 107 Reykjavík, Iceland
| | - Hannu Häkkinen
- Departments of Physics and Chemistry, Nanoscience Center, University of Jyväskylä, FI-40014 Jyväskylä, Finland
| | - Jakub Fojt
- Department of Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
| | - Jiban Kangsabanik
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Joachim Sødequist
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Jouko Lehtomäki
- Department of Applied Physics, Aalto University, P.O. Box 11100, 00076 Aalto, Finland
| | - Julian Heske
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Jussi Enkovaara
- CSC-IT Center for Science Ltd., P.O. Box 405, FI-02101 Espoo, Finland
| | - Kirsten Trøstrup Winther
- SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Marcin Dulak
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Marko M Melander
- Department of Chemistry, Nanoscience Center, University of Jyväskylä, FI-40014 Jyväskylä, Finland
| | - Martin Ovesen
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Martti Louhivuori
- CSC-IT Center for Science Ltd., P.O. Box 405, FI-02101 Espoo, Finland
| | - Michael Walter
- FIT Freiburg Centre for Interactive Materials and Bioinspired Technologies, University of Freiburg, Georges-Köhler-Allee 105, 79110 Freiburg, Germany
| | - Morten Gjerding
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Olga Lopez-Acevedo
- Biophysics of Tropical Diseases, Max Planck Tandem Group, University of Antioquia UdeA, 050010 Medellin, Colombia
| | - Paul Erhart
- Department of Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
| | - Robert Warmbier
- School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand, 1 Jan Smuts Avenue, 2001 Johannesburg, South Africa
| | - Rolf Würdemann
- Freiburger Materialforschungszentrum, Universität Freiburg, Stefan-Meier-Straße 21, D-79104 Freiburg, Germany
| | - Sami Kaappa
- Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland
| | - Simone Latini
- Nanomade, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Tara Maria Boland
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Thomas Bligaard
- Department of Energy Conversion and Storage, Technical University of Denmark, DK-2800 Lyngby, Denmark
| | - Thorbjørn Skovhus
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Toma Susi
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Tristan Maxson
- Department of Chemical and Biological Engineering, The University of Alabama, Tuscaloosa, Alabama 35487, USA
| | - Tuomas Rossi
- CSC-IT Center for Science Ltd., P.O. Box 405, FI-02101 Espoo, Finland
| | - Xi Chen
- School of Physical Science and Technology, Lanzhou University, Lanzhou, Gansu 730000, China
| | | | - Jakob Schiøtz
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Thomas Olsen
- CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
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4
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Lehtola S. A call to arms: Making the case for more reusable libraries. J Chem Phys 2023; 159:180901. [PMID: 37947507 DOI: 10.1063/5.0175165] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2023] [Accepted: 10/23/2023] [Indexed: 11/12/2023] Open
Abstract
The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of computational algorithms, computation has risen as the third cornerstone of science, joining theory and experiment on an equal footing. Computation has become an essential part of modern science, amending experiment by enabling accurate comparison of complicated theories to sophisticated experiments, as well as guiding by triage both the design and targets of experiments and the development of novel theories and computational methods. Like experiment, computation relies on continued investment in infrastructure: it requires both hardware (the physical computer on which the calculation is run) as well as software (the source code of the programs that performs the wanted simulations). In this Perspective, I discuss present-day challenges on the software side in computational chemistry, which arise from the fast-paced development of algorithms, programming models, as well as hardware. I argue that many of these challenges could be solved with reusable open source libraries, which are a public good, enhance the reproducibility of science, and accelerate the development and availability of state-of-the-art methods and improved software.
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Affiliation(s)
- Susi Lehtola
- Department of Chemistry, University of Helsinki, P.O. Box 55, FI-00014 Helsinki, Finland
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5
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Lehtola S, Marques MAL. Reproducibility of density functional approximations: How new functionals should be reported. J Chem Phys 2023; 159:114116. [PMID: 37725491 DOI: 10.1063/5.0167763] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2023] [Accepted: 08/28/2023] [Indexed: 09/21/2023] Open
Abstract
Density functional theory is the workhorse of chemistry and materials science, and novel density functional approximations are published every year. To become available in program packages, the novel density functional approximations (DFAs) need to be (re)implemented. However, according to our experience as developers of Libxc [Lehtola et al., SoftwareX 7, 1 (2018)], a constant problem in this task is verification due to the lack of reliable reference data. As we discuss in this work, this lack has led to several non-equivalent implementations of functionals such as Becke-Perdew 1986, Perdew-Wang 1991, Perdew-Burke-Ernzerhof, and Becke's three-parameter hybrid functional with Lee-Yang-Parr correlation across various program packages, yielding different total energies. Through careful verification, we have also found many issues with incorrect functional forms in recent DFAs. The goal of this work is to ensure the reproducibility of DFAs. DFAs must be verifiable in order to prevent the reappearance of the above-mentioned errors and incompatibilities. A common framework for verification and testing is, therefore, needed. We suggest several ways in which reference energies can be produced with free and open source software, either with non-self-consistent calculations with tabulated atomic densities or via self-consistent calculations with various program packages. The employed numerical parameters-especially the quadrature grid-need to be converged to guarantee a ≲0.1 μEh precision in the total energy, which is nowadays routinely achievable in fully numerical calculations. Moreover, as such sub-μEh level agreement can only be achieved when fully equivalent implementations of the DFA are used, the source code of the reference implementation should also be made available in any publication describing a new DFA.
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Affiliation(s)
- Susi Lehtola
- Molecular Sciences Software Institute, Blacksburg, Virginia 24061, USA
- Department of Chemistry, University of Helsinki, P.O. Box 55, FI-00014 Helsinki, Finland
| | - Miguel A L Marques
- Research Center Future Energy Materials and Systems of the University Alliance Ruhr, Faculty of Mechanical Engineering, Ruhr University Bochum, Universitätsstraße 150, D-44801 Bochum, Germany
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6
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Lehtola S. Atomic Electronic Structure Calculations with Hermite Interpolating Polynomials. J Phys Chem A 2023; 127:4180-4193. [PMID: 37129275 PMCID: PMC10184118 DOI: 10.1021/acs.jpca.3c00729] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We have recently described the implementation of atomic electronic structure calculations within the finite element method with numerical radial basis functions of the form χμ(r) = r-1Bμ(r), where high-order Lagrange interpolating polynomials (LIPs) were used as the shape functions Bμ(r). In this work, we discuss how χμ(r) can be evaluated in a stable manner at small r and also revisit the choice of the shape functions Bμ(r). Three kinds of shape functions are considered: in addition to the C0 continuous LIPs, we consider the analytical implementation of first-order Hermite interpolating polynomials (HIPs) that are C1 continuous, as well as numerical implementations of n-th order (Cn continuous) HIPs that are expressed in terms of an underlying high-order LIP basis. Furnished with the new implementation, we demonstrate that the first-order HIPs are reliable even with large numbers of nodes and that they also work with nonuniform element grids, affording even better results in atomic electronic structure calculations than LIPs with the same total number of basis functions. We demonstrate that discontinuities can be observed in the spin-σ local kinetic energy τσ in small LIP basis sets, while HIP basis sets do not suffer from such issues; however, either set can be used to reach the complete basis set limit with smooth τσ. Moreover, we discuss the implications of HIPs on calculations with meta-GGA functionals with a number of recent meta-GGA functionals, and we find most Minnesota functionals to be ill-behaved. We also examine the potential usefulness of the explicit control over the derivative in HIPs for forming numerical atomic orbital basis sets, but we find that confining potentials are still likely a better option.
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Affiliation(s)
- Susi Lehtola
- Molecular Sciences Software Institute, Blacksburg, Virginia 24061, United States
- Department of Chemistry, University of Helsinki, P.O. Box 55, FI-00014 Helsinki, Finland
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7
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Lehtola S. Meta-GGA Density Functional Calculations on Atoms with Spherically Symmetric Densities in the Finite Element Formalism. J Chem Theory Comput 2023; 19:2502-2517. [PMID: 37084260 PMCID: PMC10173457 DOI: 10.1021/acs.jctc.3c00183] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/22/2023]
Abstract
Density functional calculations on atoms are often used for determining accurate initial guesses as well as generating various types of pseudopotential approximations and efficient atomic-orbital basis sets for polyatomic calculations. To reach the best accuracy for these purposes, the atomic calculations should employ the same density functional as the polyatomic calculation. Atomic density functional calculations are typically carried out employing spherically symmetric densities, corresponding to the use of fractional orbital occupations. We have described their implementation for density functional approximations (DFAs) belonging to the local density approximation (LDA) and generalized gradient approximation (GGA) levels of theory as well as Hartree-Fock (HF) and range-separated exact exchange [Lehtola, S. Phys. Rev. A 2020, 101, 012516]. In this work, we describe the extension to meta-GGA functionals using the generalized Kohn-Sham scheme, in which the energy is minimized with respect to the orbitals, which in turn are expanded in the finite element formalism with high-order numerical basis functions. Furnished with the new implementation, we continue our recent work on the numerical well-behavedness of recent meta-GGA functionals [Lehtola, S.; Marques, M. A. L. J. Chem. Phys. 2022, 157, 174114]. We pursue complete basis set (CBS) limit energies for recent density functionals and find many to be ill-behaved for the Li and Na atoms. We report basis set truncation errors (BSTEs) of some commonly used Gaussian basis sets for these density functionals and find the BSTEs to be strongly functional dependent. We also discuss the importance of density thresholding in DFAs and find that all of the functionals studied in this work yield total energies converged to 0.1 μEh when densities smaller than 10-11a0-3 are screened out.
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Affiliation(s)
- Susi Lehtola
- Molecular Sciences Software Institute, Blacksburg, Virginia 24061, United States
- Department of Chemistry, University of Helsinki, P.O. Box 55, FI-00014 Helsinki, Finland
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8
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Spadetto E, Philipsen PHT, Förster A, Visscher L. Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy. J Chem Theory Comput 2023; 19:1499-1516. [PMID: 36787494 PMCID: PMC10018742 DOI: 10.1021/acs.jctc.2c01201] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023]
Abstract
Pair atomic density fitting (PADF) has been identified as a promising strategy to reduce the scaling with system size of quantum chemical methods for the calculation of the correlation energy like the direct random-phase approximation (RPA) or second-order Møller-Plesset perturbation theory (MP2). PADF can however introduce large errors in correlation energies as the two-electron interaction energy is not guaranteed to be bounded from below. This issue can be partially alleviated by using very large fit sets, but this comes at the price of reduced efficiency and having to deal with near-linear dependencies in the fit set. One posibility is to use global density fitting (DF), but in this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favorable scaling of PADF. We first regularize the Fock matrix by projecting out parts of the basis set which gives rise to orbital products that are hard to describe by PADF. After having thus obtained a reliable self-consistent field solution, we then also apply this projector to the orbital coefficient matrix to improve the precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new approach in a numerical atomic orbital framework using Slater type orbitals (STO) and correlation consistent Gaussian type basis sets up to quintuple-ζ quality for systems with more than 200 atoms. For the small and medium systems in the S66 database we show the maximum deviation of PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference results to be 0.07 and 0.14 kcal/mol, respectively. When the new projector method is used, the errors only slightly increase for large molecules and also when moderately sized fit sets are used the resulting errors are well under control. Finally, we demonstrate the computational efficiency of our algorithm by calculating the interaction energies of large, non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory.
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Affiliation(s)
- Edoardo Spadetto
- Software for Chemistry and Materials NV, NL-1081HV Amsterdam, The Netherlands
| | | | - Arno Förster
- Software for Chemistry and Materials NV, NL-1081HV Amsterdam, The Netherlands.,Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
| | - Lucas Visscher
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
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9
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Tan JA, Lao KU. Generating accurate density matrices on the tangent space of a Grassmann manifold. J Chem Phys 2023; 158:051101. [PMID: 36754784 DOI: 10.1063/5.0137775] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023] Open
Abstract
Interpolating a density matrix from a set of known density matrices is not a trivial task. This is because a linear combination of density matrices does not necessarily correspond to another density matrix. In this Communication, density matrices are examined as objects of a Grassmann manifold. Although this manifold is not a vector space, its tangent space is a vector space. As a result, one can map the density matrices on this manifold to their corresponding vectors in the tangent space and then perform interpolations on that tangent space. The resulting interpolated vector can be mapped back to the Grassmann manifold, which can then be utilized (1) as an optimal initial guess for a self-consistent field (SCF) calculation or (2) to derive energy directly without time-consuming SCF iterations. Such a promising approach is denoted as Grassmann interpolation (G-Int). The hydrogen molecule has been used to illustrate that the described interpolated method in this work preserves the essential attributes of a density matrix. For phosphorus mononitride and ferrocene, it was demonstrated numerically that reference points for the definition of the corresponding tangent spaces can be chosen arbitrarily. In addition, the interpolated density matrices provide a superior and essentially converged initial guess for an SCF calculation to make the SCF procedure itself unnecessary. Finally, this accurate, efficient, robust, and systematically improved G-Int strategy has been used for the first time to generate highly accurate potential energy surfaces with fine details for the difficult case, ferrocene.
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Affiliation(s)
- Jake A Tan
- Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284, USA
| | - Ka Un Lao
- Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284, USA
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10
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Kara M, Özdoğan T, Orbay M. Analysis of research trends on the investigation of molecular integrals over Slater type orbitals. ADVANCES IN QUANTUM CHEMISTRY 2023. [DOI: 10.1016/bs.aiq.2023.02.005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/28/2023]
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11
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Hesselmann A, Werner HJ, Knowles PJ. Thermochemical evaluation of adaptive and fixed density functional theory quadrature schemes. J Chem Phys 2022; 157:234106. [PMID: 36550055 DOI: 10.1063/5.0119622] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
A systematic study is made of the accuracy and efficiency of a number of existing quadrature schemes for molecular Kohn-Sham Density-Functional Theory (DFT) using 408 molecules and 254 chemical reactions. Included are the fixed SG-x (x = 0-3) grids of Gill et al., Dasgupta, and Herbert, the 3-zone grids of Treutler and Ahlrichs, a fixed five-zone grid implemented in Molpro, and a new adaptive grid scheme. While all methods provide a systematic reduction of errors upon extension of the grid sizes, significant differences are observed in the accuracies for similar grid sizes with various approaches. For the tests in this work, the SG-x fixed grids are less suitable to achieve high accuracies in the DFT integration, while our new adaptive grid performed best among the schemes studied in this work. The extra computational time to generate the adaptive grid scales linearly with molecular size and is negligible compared with the time needed for the self-consistent field iterations for large molecules. A comparison of the grid accuracies using various density functionals shows that meta-GGA functionals need larger integration grids than GGA functionals to reach the same degree of accuracy, confirming previous investigations of the numerical stability of meta-GGA functionals. On the other hand, the grid integration errors are almost independent of the basis set, and the basis set errors are mostly much larger than the errors caused by the numerical integrations, even when using the smallest grids tested in this work.
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Affiliation(s)
- Andreas Hesselmann
- Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany
| | - Hans-Joachim Werner
- Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany
| | - Peter J Knowles
- School of Chemistry, Cardiff University, Main Building, Park Place, Cardiff CF10 3AT, United Kingdom
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12
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Shmilovich K, Willmott D, Batalov I, Kornbluth M, Mailoa J, Kolter JZ. Orbital Mixer: Using Atomic Orbital Features for Basis-Dependent Prediction of Molecular Wavefunctions. J Chem Theory Comput 2022; 18:6021-6030. [PMID: 36122312 DOI: 10.1021/acs.jctc.2c00555] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Leveraging ab initio data at scale has enabled the development of machine learning models capable of extremely accurate and fast molecular property prediction. A central paradigm of many previous studies focuses on generating predictions for only a fixed set of properties. Recent lines of research instead aim to explicitly learn the electronic structure via molecular wavefunctions, from which other quantum chemical properties can be directly derived. While previous methods generate predictions as a function of only the atomic configuration, in this work we present an alternate approach that directly purposes basis-dependent information to predict molecular electronic structure. Our model, Orbital Mixer, is composed entirely of multi-layer perceptrons (MLPs) using MLP-Mixer layers within a simple, intuitive, and scalable architecture that achieves competitive Hamiltonian and molecular orbital energy and coefficient prediction accuracies compared to the state-of-the-art.
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Affiliation(s)
- Kirill Shmilovich
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States
| | - Devin Willmott
- Bosch Center for Artificial Intelligence, Pittsburgh, Pennsylvania 15222, United States
| | - Ivan Batalov
- Bosch Center for Artificial Intelligence, Pittsburgh, Pennsylvania 15222, United States
| | - Mordechai Kornbluth
- Bosch Research and Technology Center, Cambridge, Massachusetts 02139, United States
| | - Jonathan Mailoa
- Tencent Quantum Laboratory, Shenzhen, Guangdong 518057, China
| | - J Zico Kolter
- Bosch Center for Artificial Intelligence, Pittsburgh, Pennsylvania 15222, United States.,Carnegie Mellon University, Pittsburgh, Pennsylvania 15222, United States
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13
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Lehtola S, Karttunen AJ. Free and open source software for computational chemistry education. WIRES COMPUTATIONAL MOLECULAR SCIENCE 2022. [DOI: 10.1002/wcms.1610] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/04/2023]
Affiliation(s)
- Susi Lehtola
- Molecular Sciences Software Institute Blacksburg Virginia USA
| | - Antti J. Karttunen
- Department of Chemistry and Materials Science Aalto University Espoo Finland
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14
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Kasim MF, Lehtola S, Vinko SM. DQC: A Python program package for differentiable quantum chemistry. J Chem Phys 2022; 156:084801. [DOI: 10.1063/5.0076202] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Automatic differentiation represents a paradigm shift in scientific programming, where evaluating both functions and their derivatives is required for most applications. By removing the need to explicitly derive expressions for gradients, development times can be shortened and calculations can be simplified. For these reasons, automatic differentiation has fueled the rapid growth of a variety of sophisticated machine learning techniques over the past decade, but is now also increasingly showing its value to support ab initio simulations of quantum systems and enhance computational quantum chemistry. Here, we present an open-source differentiable quantum chemistry simulation code and explore applications facilitated by automatic differentiation: (1) calculating molecular perturbation properties, (2) reoptimizing a basis set for hydrocarbons, (3) checking the stability of self-consistent field wave functions, and (4) predicting molecular properties via alchemical perturbations.
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Affiliation(s)
- Muhammad F. Kasim
- Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
| | - Susi Lehtola
- Molecular Sciences Software Institute, Blacksburg, Virginia 24061, USA
| | - Sam M. Vinko
- Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
- Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom
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15
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Trepte K, Schwalbe S, Liebing S, Schulze WT, Kortus J, Myneni H, Ivanov AV, Lehtola S. Chemical bonding theories as guides for self-interaction corrected solutions: Multiple local minima and symmetry breaking. J Chem Phys 2021; 155:224109. [PMID: 34911315 DOI: 10.1063/5.0071796] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Fermi-Löwdin orbitals (FLOs) are a special set of localized orbitals, which have become commonly used in combination with the Perdew-Zunger self-interaction correction (SIC) in the FLO-SIC method. The FLOs are obtained for a set of occupied orbitals by specifying a classical position for each electron. These positions are known as Fermi-orbital descriptors (FODs), and they have a clear relation to chemical bonding. In this study, we show how FLOs and FODs can be used to initialize, interpret, and justify SIC solutions in a common chemical picture, both within FLO-SIC and in traditional variational SIC, and to locate distinct local minima in either of these approaches. We demonstrate that FLOs based on Lewis theory lead to symmetry breaking for benzene-the electron density is found to break symmetry already at the symmetric molecular structure-while ones from Linnett's double-quartet theory reproduce symmetric electron densities and molecular geometries. Introducing a benchmark set of 16 planar cyclic molecules, we show that using Lewis theory as the starting point can lead to artifactual dipole moments of up to 1 D, while Linnett SIC dipole moments are in better agreement with experimental values. We suggest using the dipole moment as a diagnostic of symmetry breaking in SIC and monitoring it in all SIC calculations. We show that Linnett structures can often be seen as superpositions of Lewis structures and propose Linnett structures as a simple way to describe aromatic systems in SIC with reduced symmetry breaking. The role of hovering FODs is also briefly discussed.
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Affiliation(s)
- Kai Trepte
- SUNCAT Center for Interface Science and Catalysis, Stanford University, Menlo Park, California 94025, USA
| | - Sebastian Schwalbe
- Institute of Theoretical Physics, TU Bergakademie Freiberg, D-09599 Freiberg, Germany
| | - Simon Liebing
- Joint Institute for Nuclear Research Dubna, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Russia
| | - Wanja T Schulze
- Institute of Theoretical Physics, TU Bergakademie Freiberg, D-09599 Freiberg, Germany
| | - Jens Kortus
- Institute of Theoretical Physics, TU Bergakademie Freiberg, D-09599 Freiberg, Germany
| | - Hemanadhan Myneni
- Science Institute and Faculty of Physical Sciences, VR-III, University of Iceland, 107 Reykjavík, Iceland
| | - Aleksei V Ivanov
- Science Institute and Faculty of Physical Sciences, VR-III, University of Iceland, 107 Reykjavík, Iceland
| | - Susi Lehtola
- Molecular Sciences Software Institute, Blacksburg, Virginia 24061, USA
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16
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Lehtola S. Straightforward and Accurate Automatic Auxiliary Basis Set Generation for Molecular Calculations with Atomic Orbital Basis Sets. J Chem Theory Comput 2021; 17:6886-6900. [PMID: 34614349 DOI: 10.1021/acs.jctc.1c00607] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Density fitting (DF), also known as the resolution of the identity (RI), is a widely used technique in quantum chemical calculations with various types of atomic basis sets─Gaussian-type orbitals, Slater-type orbitals, as well as numerical atomic orbitals─to speed up density functional, Hartree-Fock (HF), and post-HF calculations. Traditionally, custom auxiliary basis sets are hand-optimized for each orbital basis set; however, some automatic schemes have also been suggested. In this work, we propose a simple yet numerically stable automated scheme for forming auxiliary basis sets with the help of a pivoted Cholesky decomposition, which is applicable to any type of atomic basis function. We exemplify the scheme with proof-of-concept calculations with Gaussian basis sets and show that the proposed approach leads to negligible DF/RI errors in HF and second-order Møller-Plesset (MP2) total energies of the non-multireference part of the W4-17 test set when used with orbital basis sets of at least polarized triple-ζ quality. The results are promising for future applications employing Slater-type orbitals or numerical atomic orbitals, as well as schemes based on the use of local fitting approaches. Global fitting approaches can also be used, in which case the high-angular-momentum functions produced by the present scheme can be truncated to minimize the computational cost.
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Affiliation(s)
- Susi Lehtola
- Molecular Sciences Software Institute, Blacksburg, Virginia 24061, United States
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17
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Accurate acid dissociation constant (pK a) calculation for the sulfachloropyridazine and similar molecules. J Mol Model 2021; 27:233. [PMID: 34324066 DOI: 10.1007/s00894-021-04851-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2021] [Accepted: 07/05/2021] [Indexed: 10/20/2022]
Abstract
Accurate calculation of the acid dissociation constant (pKa) has fundamental importance for the description of molecular systems with pharmacological activities. The search for a more appropriate procedure for its determination is always welcome and has aroused increasing interest from the scientific community. In this sense, this work presents a computational study involving the combination of ten DFT functionals (M062X, M06L, B3LYP, BLYP, PBEPBE, BP86, LC-BLYP, SPBE, CAM-B3LYP, LC-PBEPBE) and HF method, eight basis set functions (6-311G, 6-311 + G, 6-311G(d,p), 6-311 + G(d,p), 6-311+ +G(d,p), 6-311(2d,2p), 6-311+ +G(2d,2p), and aug-cc-pVDZ), and three solvation models (SMD, PCM, and CPCM) for an accurate sulfachloropyridazine (SCR) pKa determination. It was found that the smallest deviation (0.02 unit of pKa) between the current study and experimental result was achieved with the BLYP/6-311 + G(d,p)/PCM combination. Therefore, this combination was extended to calculate the pKa of six SCR similar molecules selected through the eletroshape similarity method. For all these molecules, the difference between the obtained results and experimental data ranged between 0.14 and 0.69 units of pKa. This feature suggests that the obtained combination can determine pKa with experimental precision for complexes that are formed by sulfonamide functional group (SO2NHR). Graphical Abstract A computational study involving the combination of different levels of theory, basis sets and solvation models for an accurate sulfanamide pKa determination.
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18
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Lehtola S, Marques MAL. Meta-Local Density Functionals: A New Rung on Jacob's Ladder. J Chem Theory Comput 2021; 17:943-948. [PMID: 33502183 PMCID: PMC8023657 DOI: 10.1021/acs.jctc.0c01147] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
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 nσ, 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 nσavg(r) = nσ1-x(r)ñσx(r) of the local spin density nσ(r) and the spin density ñσ(r) of a HEG that has the local kinetic energy density τσ(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 [ 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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Affiliation(s)
- Susi Lehtola
- Department of Chemistry, University of Helsinki, P.O. Box 55 (A.I. Virtasen aukio 1), FI-00014 University of Helsinki, Finland
| | - Miguel A L Marques
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle, Saale, Germany
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19
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Förster A, Visscher L. Low-Order Scaling G0W0 by Pair Atomic Density Fitting. J Chem Theory Comput 2020; 16:7381-7399. [PMID: 33174743 PMCID: PMC7726916 DOI: 10.1021/acs.jctc.0c00693] [Citation(s) in RCA: 33] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Indexed: 12/18/2022]
Abstract
We derive a low-scaling G0W0 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 G0W0 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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Affiliation(s)
- Arno Förster
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
| | - Lucas Visscher
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
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20
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Levi G, Ivanov AV, Jónsson H. Variational Density Functional Calculations of Excited States via Direct Optimization. J Chem Theory Comput 2020; 16:6968-6982. [DOI: 10.1021/acs.jctc.0c00597] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Affiliation(s)
- Gianluca Levi
- Science Institute and Faculty of Physical Sciences, University of Iceland, 107 Reykjavík, Iceland
| | - Aleksei V. Ivanov
- Science Institute and Faculty of Physical Sciences, University of Iceland, 107 Reykjavík, Iceland
- Saint Petersburg State University, 199034 Saint Petersburg, Russia
| | - Hannes Jónsson
- Science Institute and Faculty of Physical Sciences, University of Iceland, 107 Reykjavík, Iceland
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21
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Schwalbe S, Fiedler L, Kraus J, Kortus J, Trepte K, Lehtola S. PyFLOSIC: Python-based Fermi–Löwdin orbital self-interaction correction. J Chem Phys 2020; 153:084104. [DOI: 10.1063/5.0012519] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sebastian Schwalbe
- Institute of Theoretical Physics, TU Bergakademie Freiberg, Leipziger Str. 23, D-09599 Freiberg, Germany
| | - Lenz Fiedler
- Institute of Theoretical Physics, TU Bergakademie Freiberg, Leipziger Str. 23, D-09599 Freiberg, Germany
| | - Jakob Kraus
- Institute of Theoretical Physics, TU Bergakademie Freiberg, Leipziger Str. 23, D-09599 Freiberg, Germany
| | - Jens Kortus
- Institute of Theoretical Physics, TU Bergakademie Freiberg, Leipziger Str. 23, D-09599 Freiberg, Germany
| | - Kai Trepte
- Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA
| | - Susi Lehtola
- Department of Chemistry, University of Helsinki, P.O. Box 55 (A. I. Virtasen Aukio 1), FI-00014 University of Helsinki, Finland
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22
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Lehtola S, Visscher L, Engel E. Efficient implementation of the superposition of atomic potentials initial guess for electronic structure calculations in Gaussian basis sets. J Chem Phys 2020; 152:144105. [DOI: 10.1063/5.0004046] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Susi Lehtola
- Department of Chemistry, University of Helsinki, P.O. Box 55 (A. I. Virtasen aukio 1), FI-00014 Helsinki, Finland
| | - Lucas Visscher
- Division of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
| | - Eberhard Engel
- Center for Scientific Computing, J. W. Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany
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23
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Levi G, Ivanov AV, Jónsson H. Variational calculations of excited states via direct optimization of the orbitals in DFT. Faraday Discuss 2020; 224:448-466. [DOI: 10.1039/d0fd00064g] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Abstract
A direct optimization method for obtaining excited electronic states using density functionals is presented.
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Affiliation(s)
- Gianluca Levi
- Science Institute and Faculty of Physical Sciences
- University of Iceland
- Iceland
| | - Aleksei V. Ivanov
- Science Institute and Faculty of Physical Sciences
- University of Iceland
- Iceland
- Saint Petersburg State University
- 199034 Saint Petersburg
| | - Hannes Jónsson
- Science Institute and Faculty of Physical Sciences
- University of Iceland
- Iceland
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