Self-consistent implementation of meta-gga exchange-correlation functionals within the ONETEP linear-scaling DFT code
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1 Self-consistent implementation of meta-gga exchange-correlation functionals within the ONETEP linear-scaling DFT code James C. Womack, Chris-Kriton Skylaris Chemistry, Faculty of Natural & Environmental Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ, UK RSC Theoretical Chemistry Group Conference, Nottingham, 2016
2 Overview Background & Motivation Theory Linear-scaling DFT ONETEP formalism Density functional approximations Meta-GGA functionals Approaches to self-consistency Why? Implementation General meta-gga framework in ONETEP Results & Conclusions Numerical validation Demonstration of linear-scaling Specific functionals implemented 2
3 Linear-scaling density functional theory Orthogonalize KS orbitals extending over entire system Conventional Kohn-Sham DFT Cubically scaling computational cost Linear-scaling density matrix DFT The density matrix is nearsighted For insulators:* Exploit this to obtain linear scaling computational cost *see e.g. S. Ismail-Beigi and T.A. Arias, Phys. Rev. Lett. 82, 2127 (1999) 3
4 Approximate exchange-correlation functionals No practical exact form is known Ideally, these should be Accurate Computationally efficient Hierarchical series New ingredients on each rung double-hybrid non-local Jacob's ladder of approximations Heaven of chemical accuracy Approximations are required hybrid meta-gga (semi-)local GGA Higher rungs are more flexible Which is the best tool for the job? LDA Hartree world J.P. Perdew and K. Schmidt, AIP Conf. Proc. 577, 1 (2001) 4
5 Motivation: The meta-gga Goldilocks zone Non-local ingredients (e.g. exact exchange) pose a challenge to linear scaling* Increasing accuracy Meta-GGAs sit in a Goldilocks zone Well-balanced cost vs. accuracy Simple semi-local form Add kinetic energy density, τ More flexible functional forms Greater constraint satisfaction** Modern meta-ggas offer impressive accuracy e.g. B97M-V, M15-L, SCAN... Goldilocks zone * Linear scaling exact exchange has been achieved in ONETEP but with a relatively large prefactor: J. Dziedzic, Q. Hill, and C.-K. Skylaris, J. Chem. Phys. 139, (2013) Decreasing computational cost ** For more on constraint satifaction: J.P. Perdew, A. Ruzsinszky, J. Tao, V.N. Staroverov, G.E. Scuseria, and G.I. Csonka, J. Chem. Phys. 123, (2005) 5
6 Order-N Electronic Total Energy Package NGWFs Non-orthogonal generalized Wannier functions Strictly localized orbitals Self-consistently optimized Periodic cardinal sinc psinc basis Density kernel Sparse matrix Truncated above cut-off length, i.e. C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, and M.C. Payne, J. Chem. Phys. 122, (2005) 6
7 Order-N Electronic Total Energy Package Total energy minimization Via two nested loops, i.e. Initial guess Optimize K Converged? Yes No Optimize NGWFs Psinc basis No Converged? Orthogonal Bandwidth-limited delta functions on a regular real-space grid Finite sum over plane waves, i.e. Yes Energy minimized 7
8 Self-consistent meta-gga evaluation Unknown dependence of τ on the density prevents direct functional derivative evaluation There are three main approaches for overcoming this: 1. Give up! 2. Optimized effective potential Perform meta-gga calculations nonself-consistently Obtain a local, multiplicative potential which satisfies Use orbitals optimized using a non-τdependent functional (LDA, GGA) Yields surprisingly good results Example: J.P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. 82, 2544 (1999). by solving the OEP integral equation Formidable mathematical machinery (even with approximations) Example: A.V. Arbuznikov and M. Kaupp, Chem. Phys. Lett. 381, 495 (2003). 8
9 3. FDO approximation Functional Derivatives (of τ-dependent functionals) with respect to the Orbitals* Use the functional derivative chain rule to obtain The functional derivative allows us to derive the following relationship * AKA NNH approach after the creators: R. Neumann, R.H. Nobes, and N.C. Handy, Mol. Phys. 87, 1 (1996) This functional derivative is approximate, since it neglects the dependence of the KS orbitals on each other via the density, i.e. For a detailed account, see: F. Zahariev, S.S. Leang, and M.S. Gordon, J. Chem. Phys. 138, (2013) 9
10 3. FDO approximation We know the functional dependence of τ on the KS orbitals FDO Non-multiplicative, orbital-specific potential and can therefore evaluate and obtain our FDO XC potential using This approach has the advantages of Theoretical simplicity Computational simplicity Wide use in other code (i.e. well-tested) A.V. Arbuznikov and M. Kaupp, Chem. Phys. Lett. 381, 495 (2003). 10
11 Implementation overview Kinetic energy density Exchange-correlation potential Self-consistent meta-gga functionals in ONETEP Integral over τ-dependent potential Energy gradients Functional forms 11
12 Kinetic energy density We require τ in terms of the NGWFs and density kernel, i.e. In practice, we evaluate τ via a multi-step process: 1 Evaluate intermediate quantity, summing over NGWFs which overlap with α 2 Form part of τ local to each α NGWF 3 Deposit each α-component in simulation cell Based on the approach used to evaluate the charge density in ONETEP: C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, and M.C. Payne, Phys. Stat. Sol. (B) 243, 973 (2006) 12
13 Exchange-correlation potential We require an XC potential in terms of NGWFs, not KS orbitals: FDO Expand KS orbital in terms of NGWFs Per-NGWF exchange-correlation potential 13
14 Integral over τ-dependent potential Hamiltonian matrix elements contain integrals over the τ-dependent potential: Gradient of this function avoided using integration-by-parts Form originally used by NNH and in later implementations No problem for ONETEP! Apply divergence to vector field in reciprocal space Form used in ONETEP Integrand is zero outside of localization region of α 14
15 Energy gradients Gradient expressions are required for ONETEP's energy minimization scheme. Only minor changes and additions to the existing expressions are needed. New integral type Q matrix depends on New τ-dependent potential term 15
16 Functionals PKZB An early meta-gga (1999) Modern meta-gga (2015) 1 empirical parameter Combinatorially designed form Evolved into TPSS (2003) Empirically fitted to very large data set Simple to implement Based on PBE, but with τ Incorporates non-local VV10 functional Widely available B97M-V Stable implementations to compare and test against Excellent accuracy Comparable to popular hybrid functionals Broadly applicable Inconsistent accuracy Superseded by modern meta-ggas J.P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. 82, 2544 (1999) Available in ONETEP Code provided by N. Mardirossian N. Mardirossian and M. Head-Gordon, J. Chem. Phys. 142, (2015) 16
17 Aside: How good is B97M-V? Thermochemical data Non-covalent data 2460 data points RMSD Hybrid M06-2X 3.21 ωb97x-v 3.60 ωb97x-d 3.61 B97M-V 3.93 B97-D M11 RMSD Hybrid B97M-V 0.22 ωb97x-v 0.32 M06-L 0.42 B97-D ωb97x-d M M M B3LYP-D M06-L TPSS-D VV PBE-D PBE-D VV Zero Zero Performs well against hybrid functionals! M06-2X NLC is not sufficient for good performance B3LYP-D3 in either set TPSS-D aqz basis set atz basis set RMSDs in kcal/mol for thermochemical and non-covalent datasets (combined training and test data). Based on Table V of N. Mardirossian and M. Head-Gordon, J. Chem. Phys. 142, (2015). 17
18 Results I: Numerical testing Water dimer Å Comparison of relative energies for B97M-V QChem (Gaussian basis, BSSE corrected) ONETEP (NGWFs/psincs) B97M-rV* Complicated by several possible sources of error Pseudopotential vs. all-electron Periodic vs. open boundary conditions Binding energy Program Basis PBE B97M-rV E(ONETEP) E(QChem) PBE B97M-rV - - QChem aug-pc ONETEP 900eV / 8 bohr ONETEP 900eV / 12 bohr ONETEP 900eV / 15 bohr ONETEP 1200eV / 8 bohr ONETEP 1200eV / 12 bohr ONETEP 1200eV / 15 bohr Water dimer binding energies in kcal/mol. QChem results provided by N. Mardirossian (UC Berkeley). * rv refers to revised VV10: R. Sabatini, T. Gorni, and S. de Gironcoli, Phys. Rev. B 87, (2013) 18
19 Results I: Numerical testing Water dimer The 8 bohr results seem too good to be true! Further ONETEP calculation details: Why are the results worse for larger NGWF radii? Use ONETEP's extended NGWF functionality. Possibly a favourable cancellation of errors (unfortunately). Basis 24Å cubic simulation cell Norm-conserving pseudopotentials* E(ONETEP) E(QChem) E(ONETEP,local) E(ONETEP,extended) PBE PBE B97M-rV B97M-rV 900eV / 8.0 bohr eV / 12.0 bohr eV / 15.0 bohr eV / Extended eV / 8.0 bohr eV / 12.0 bohr eV / 15.0 bohr eV / Extended Energy differences in kcal/mol. QChem results provide by N. Mardirossian (UC Berkeley). *Rappe-Bennett pseudopotential library: 19
20 Results II: Computational efficiency atom structure Iridis 4 supercomputer 128 MPI processes 4 OpenMP threads Structure provided by authors of J.T. Berryman, S.E. Radford, and S.A. Harris, Biophys. J. 97, 1 (2009) 20
21 Conclusions Self-consistent meta-gga functional evaluation in ONETEP Theoretical and computational framework implemented and tested Further functional forms can easily be added Meta-GGAs can be applied to very large systems Provisional numerical agreement with other implementations Further testing and validation is ongoing Linear-scaling computational cost Self-consistent evaluation of meta-gga functionals Demonstrated for systems with 1000s of atoms 21
22 The future Extensions to the framework Support for projector augmented waves (PAW) Support for hybrid meta-gga functionals using existing exact exchange in ONETEP* Additional meta-gga functional forms TPSS, SCAN, M15-L are of particular interest Public availability Self-consistent meta-gga support will be made available in a future version of ONETEP *J. Dziedzic, Q. Hill, and C.-K. Skylaris, J. Chem. Phys. 139, (2013) 22
23 Acknowledgements People Martin Head-Gordon (Berkeley) Narbe Mardirossian (Berkeley) Jacek Dziedzic (Southampton) Compute resources Funding University of Southampton IRIDIS HPC facility Engineering and Physical Sciences Research Council University of Southampton 23
24 Original artwork by Lauren Womack
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