A self-interaction free non-local correlation functional derived from exact criteria
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1 A self-interaction free non-local correlation functional derived from exact criteria Tobias Schmidt and Stephan Kümmel Theoretical Physics IV University of Bayreuth Germany - International Workshop DFT meets QIT, Araraquara, Brazil,
2 What to gain from a Kohn-Sham calculation? Choose/develop approximation for 2
3 What to gain from a Kohn-Sham calculation? Choose/develop approximation for structures/geometries energetics of binding/dissociation 2
4 What to gain from a Kohn-Sham calculation? Choose/develop approximation for structures/geometries linear response theory energetics of binding/dissociation prediction of photoemission spectra/ density of states band-structure interpretation strictly physical: 2
5 What to gain from a Kohn-Sham calculation? Choose/develop approximation for structures/geometries energetics of binding/dissociation linear response theory prediction of photoemission spectra/ density of states band-structure interpretation strictly physical: Satisfying description of both quantities using a single approximation? 2
6 Local vs. Non-local functionals (semi-)local (LDA/GGA) Amount of non-locality Binding? Eigenvalue? One-electron self-interaction (SI)? pure exact exchange (EXX) (Fock-exchange with KS orbitals) 3
7 Local vs. Non-local functionals (semi-)local (LDA/GGA) Amount of non-locality pure exact exchange (EXX) inherent Binding? Eigenvalue? One-electron self-interaction (SI)? free 3
8 Local vs. Non-local functionals (semi-)local (LDA/GGA) Amount of non-locality pure exact exchange (EXX) inherent Binding? Eigenvalue? One-electron self-interaction (SI)? free Global hybrid functionals Use a fixed, constant amount of non-local and (semi-)local components 3
9 Local vs. Non-local functionals (semi-)local (LDA/GGA) Amount of non-locality pure exact exchange (EXX) inherent Binding? Eigenvalue? One-electron self-interaction (SI)? free Global hybrid functionals Use a fixed, constant amount of non-local and (semi-)local components Good binding energies and structures! But: non-vanishing SI error wrong potential asymptotics eigenvalues not of satisfying quality 3
10 Local vs. Non-local functionals (semi-)local (LDA/GGA) Amount of non-locality pure exact exchange (EXX) inherent Binding? Eigenvalue? One-electron self-interaction (SI)? free Global hybrid functionals Use a fixed, constant amount of non-local and (semi-)local components Good binding energies and structures! But: non-vanishing SI error wrong potential asymptotics eigenvalues not of satisfying quality Introduce more flexibility in mixing of local and non-local components! 3
11 Constructing a local hybrid functional General ansatz: 4
12 Constructing a local hybrid functional General ansatz: Global hybrids: 4
13 Constructing a local hybrid functional General ansatz: Global hybrids: Local hybrids: 4
14 Constructing a local hybrid functional Construct a physically motivated local mixing function : Satisfy single electron case Eliminating oneelectron SI error Satisfy homogeneous electron gas limit Correct asymptotic behavior Include full EXX 5 T. Schmidt, E. Kraisler, A. Makmal, L. Kronik and S. Kümmel, J. Chem. Phys. 140, 18A510 (2014).
15 The full functional expression Bohr radius 6 T. Schmidt, E. Kraisler, A. Makmal, L. Kronik and S. Kümmel, J. Chem. Phys. 140, 18A510 (2014).
16 A new local hybrid: Testing the performance Local mixing function contains one initially undetermined parameter c (no universal constraint to determine this parameter) 7
17 A new local hybrid: Testing the performance Local mixing function contains one initially undetermined parameter c (no universal constraint to determine this parameter) How to test the functional's performance? 1. Test set: 18 small molecules and their constituent atoms 2. Compute average relative errors in dependence on the parameter c: denotes the binding energy or the via, respectively 7
18 Average relative error of the binding energy for different functionals 8
19 Optimal description with Average relative error of the binding energy for different functionals 8
20 Average relative error of the ionization potential via the highest occupied eigenvalue for different functionals 9
21 Optimal description with Average relative error of the ionization potential via the highest occupied eigenvalue for different functionals 9
22 Interpreting the results Accuracy similar to best global hybrid for dissociation energies + Improving interpretability of the highest occupied eigenvalue 10
23 Interpreting the results Accuracy similar to best global hybrid for dissociation energies + Improving interpretability of the highest occupied eigenvalue Optimal description: completely different range of values of the parameter c required 10
24 Interpreting the results Accuracy similar to best global hybrid for dissociation energies + Improving interpretability of the highest occupied eigenvalue Optimal description: completely different range of values of the parameter c required Spatially flexible mixing alone does not completely solve the duality 10
25 How can this remaining duality be explained? 1. Issue for many other functionals: global hybrids range-separated hybrids (scaled-down) SI correction Important physics is generally missed in many functionals. 11
26 How can this remaining duality be explained? 2. Detailed analysis of potential asymptotics for local hybrids in general: with All local hybrids (even though one-electron SI-free in principle), reach an incorrect potential fall-off determined by the local mixing function. 12 T. Schmidt, E. Kraisler, L. Kronik and S. Kümmel, Phys. Chem. Chem. Phys. 16, (2014).
27 Calculation of binding curves H 2 + He T. Schmidt, E. Kraisler, A. Makmal, L. Kronik and S. Kümmel, J. Chem. Phys. 140, 18A510 (2014).
28 Acknowledgements Leeor Kronik and Eli Kraisler Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth, Israel Adi Makmal Institute for Theoretical Physics, University of Innsbruck, Austria Elite Network of Bavaria Thank you for your attention! 14
29 Outlook Not only the xc potential, but also important quantities frequently used in the construction of functionals show an unexpected behavior: Single-orbital indicator function: Fig. 3: Indicator function on the numerical grid for the carbon atom. A nodal plane in the HOMO occurs along the z-axis. 15 T. Schmidt, E. Kraisler, L. Kronik and S. Kümmel, Phys. Chem. Chem. Phys., (2014).
30 Backup/Details the full local hybrid functional expression details regarding the asymptotic decay of the xc potential of local hybrids B1
31 The full functional expression with: Von Weizsäcker kinetic energy density Kohn-Sham kinetic energy density B2 T. Schmidt, E. Kraisler, A. Makmal, L. Kronik and S. Kümmel, J. Chem. Phys. 140, (2014).
32 The full functional expression Spin polarization Reduced density gradient with: Bohr radius B3 T. Schmidt, E. Kraisler, A. Makmal, L. Kronik and S. Kümmel, J. Chem. Phys. 140, (2014).
33 Details regarding the potential asymptotics intuitively since If the correct asymptotic behavior of the potential is obtained for any vanishing local mixing function, then why does a dependence on c occur for the eigenvalue? B4
34 Details regarding the potential asymptotics due to non-local evaluation of One finds analytically: The local mixing function has a direct influence on the asymptotics of the xc potential. Consequently, local hybrids do not reach the correct asymptotics, which explains the remaining duality in the description of total energies and Kohn-Sham eigenvalues. B5 T. Schmidt, E. Kraisler, L. Kronik and S. Kümmel, Phys. Chem. Chem. Phys., (2014).
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