Introduction to the OCTOPUS code
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1 Introduction to the OCTOPUS code Esa Räsänen Nanoscience Center, University of Jyväskylä, Finland
2
3 Brief history Origins in the fixed nucleus code of Bertsch and Yabana [1] and in the real space code of Rubio, Blase, and Louie [2] Major re writing by Alberto Castro and Miguel Marques in Fortran 95 and C [3,4]. [1] G. F. Bertsch and K. Yabana, Time dependent local density approximation in real time, Phys. Rev. B 54, 4484 (1996) [2] A. Rubio, X. Blase, and S. G. Louie, Ab Initio Photoabsorption Spectra and Structures of Small Semiconductor and Metal Clusters, Phys. Rev. Lett. 77, 247 (1996) [3] M. A. L. Marques, A. Castro, G. F. Bertsch, and A. Rubio, octopus: a first principles tool for excited electron ion dynamics, Comput. Phys. Commun. 151, (2003) [4] A. Castro, H. Appel, M. Oliveira, C. A. Rozzi, X. Andrade, F. Lorenzen, M.A.L. Marques, E. K. U. Gross, and A. Rubio, octopus: a tool for the application of time dependent density functional theory, Phys. Stat. Sol. (b) 243, 2465 (2006)
4 Features Theoretical base: DFT and TDDFT Focus on finite systems (periodic systems in progress) Real space grid presentation (numerical mesh) Auxiliary basis sets (e.g., atomic orbitals) used when necessary Possibility for adaptive grids and multigrids Pseudopotentials 3D / 2D / 1D Parallel scalability ~ lines of Fortran, C, and Perl Licenced under GNU developers welcome!
5 Target problems Linear response of molecules and clusters Non linear response to high intensity electromagnetic fields Ground and excited state properties of low dimensional systems Photo induced reactions of molecules (e.g., photo dissociation) Optimal control theory Electronic transport (under development)
6 Pseudopotentials density of valence electrons effects of core electrons included in the external potential Options: Troullier Martins / Hartwigsen Goedecker Hutter Real space grid Partial differential equations > infinite degrees of freedom Reduction to a finite number: Function represented by its value over a set of points Uniform or adaptive point distribution in a finite box Grid points play the role of (non variational) "basis functions"
7 Boundary conditions Functions in finite systems go to zero => functions imposed to be zero at the border of the box => box has to be large enough to contain the functions Box shape should be optimized to minimize the number of points Options: minimum box (sphere around each atom) sphere cylinder rectangle arbitrary
8 Differential operations Finite difference approach Derivative in a point calculated from a sum over neighboring points Coefficients depend on the points used (stencil) More points > more precision Stencil options: star star + off diagonal cube
9 Integration Trapezoidal rule Eigensolver Options: Lanczos method iterative conjugate gradient algorithm (default) imaginary time propagation (in development)
10 Density mixing scheme: or, alternatively, mixing of the potentials (EXX) linear mixing good for "tough" situations Broyden mixing (default) D. D. Johnson, Modified Broyden s method for accelerating convergence in self consistent calculations, Phys. Rev. B 38, (1988)
11 Parallelization In domains: each processor handles points in regions in space points in the boundaries must be copied to other nodes integrals performed locally and summed over all domains efficient and well scalable scheme In states: each processor handles a group of states efficient scheme for time propagation Current status of scalability: hundreds of processors
12 Selection of exchange correlation functionals lda_c_pz_mod (10000): LDA: Perdew & Zunger gga_x_pbe (101): GGA: Perdew, Burke & Ernzerhof gga_x_pbe_r (102): GGA: Perdew, Burke & Ernzerhof (revised) gga_x_b86 (103): GGA: Becke 86 Xalpha,beta,gamma gga_x_b86_r (104): GGA: Becke 86 Xalpha,beta,gamma reoptimized gga_x_b86_mgc (105): GGA: Becke 86 Xalpha,beta,gamma gga_x_b88 (106): GGA: Becke 88 gga_x_g96 (107): GGA: Gill 96 gga_x_pw86 (108): GGA: Perdew & Wang 86 gga_x_pw91 (109): GGA: Perdew & Wang 91 lda_c_ob_pz (11000): LDA: Ortiz & Ballone (PZ-type parametrization) gga_x_optx (110): GGA: Handy & Cohen OPTX 01 gga_x_dk87_r1 (111): GGA: depristo & Kress 87 (version R1) gga_x_dk87_r2 (112): GGA: depristo & Kress 87 (version R2) gga_x_lg93 (113): GGA: Lacks & Gordon 93 gga_x_ft97_a (114): GGA: Filatov & Thiel 97 (version A) gga_x_ft97_b (115): GGA: Filatov & Thiel 97 (version B) gga_x_pbe_sol (116): GGA: Perdew, Burke & Ernzerhof (solids) gga_x_rpbe (117): GGA: Hammer, Hansen & Norskov (PBE-like) gga_x_wc (118): GGA: Wu & Cohen gga_x_mpw91 (119): GGA: Modified form of PW91 by Adamo & Barone lda_c_pw (12000): LDA: Perdew & Wang gga_x_am05 (120): GGA: Armiento & Mattsson 05 exchange gga_x_pbea (121): GGA: Madsen 07 (PBE-like) gga_x_mpbe (122): GGA: Adamo & Barone modification to PBE gga_x_xpbe (123): GGA: xpbe reparametrization by Xu & Goddard gga_x_2d_b86_mgc (124): GGA: Becke 86 MGC for 2D systems gga_c_pbe (130000): GGA: Perdew, Burke & Ernzerhof correlation lda_c_pw_mod (13000): LDA: Perdew & Wang gga_c_lyp (131000): GGA: Lee, Yang, & Parr gga_c_p86 (132000): GGA: Perdew 86 gga_c_pbe_sol (133000): GGA: Perdew, Burke & Ernzerhof correlation gga_c_pw91 (134000): GGA: Perdew & Wang 91 gga_c_am05 (135000): GGA: Armiento & Mattsson 05 correlation gga_c_xpbe (136000): GGA: xpbe reparametrization by Xu & Goddard gga_c_lm (137000): GGA: Langreth and Mehl correlation lda_c_ob_pw (14000): LDA: Ortiz & Ballone (PW-type parametrization) lda_c_2d_amgb (15000): LDA: Attacalite et al functional for the 2D electron gas lda_c_2d_prm08 (16000): LDA: Pittalis, Rasanen & Marques correlation in 2D gga_xc_lb (160): GGA: van Leeuwen & Baerends (xc) gga_xc_hcth_93 (161): GGA: HCTH fit to 93 molecules (xc) gga_xc_hcth_120 (162): GGA: HCTH fit to 120 molecules (xc) gga_xc_hcth_147 (163): GGA: HCTH fit to 147 molecules (xc) gga_xc_hcth_407 (164): GGA: HCTH fit to 407 molecules (xc) gga_xc_edf1 (165): GGA: Empirical functional from Adamson, Gill, & Pople gga_xc_xlyp (166): GGA: Empirical functional from Xu & Goddard lda_c_vbh (17000): LDA: von Barth & Hedin lda_x_2d (19): LDA: Slater exchange lda_x (1): LDA: Slater exchange lda_c_wigner (2000): LDA: Wigner parametrization mgga_x_lta (201): MGGA: Local tau approximation of Ernzerhof & Scuseria mgga_x_tpss (202): MGGA: Perdew, Tao, Staroverov & Scuseria exchange mgga_x_m06l (203): MGGA: Zhao, Truhlar exchange mgga_x_gvt4 (204): MGGA: GVT4 from Van Voorhis and Scuseria mgga_x_tau_hcth (205): MGGA: tau-hcth from Boese and Handy lda_xc_teter93 (20): LDA: Teter 93 pade parametrization mgga_c_tpss (231000): MGGA: Perdew, Tao, Staroverov & Scuseria correlation mgga_c_vsxc (232000): MGGA: VSxc from Van Voorhis and Scuseria (correlation part) lda_c_rpa (3000): LDA: Random Phase Approximation lda_c_hl (4000): LDA: Hedin & Lundqvist hyb_gga_xc_b3pw91 (401000): Hybrid (GGA): the first hybrid by Becke hyb_gga_xc_b3lyp (402000): Hybrid (GGA): The (in)famous B3LYP hyb_gga_xc_b3p86 (403000): Hybrid (GGA): Perdew 86 hybrid similar to B3PW91 hyb_gga_xc_o3lyp (404000): Hybrid (GGA): Hybrid using the optx functional hyb_gga_xc_pbeh (406000): Hybrid (GGA): aka PBE0 or PBE1PBE hyb_gga_xc_x3lyp (411000): Hybrid (GGA): maybe the best hybrid around hyb_gga_xc_b1wc (412000): Hybrid (GGA): Becke 1-parameter mixture of WC and EXX lda_c_gl (5000): LDA: Gunnarson & Lundqvist lda_c_xalpha (6000): LDA: Slater s Xalpha lda_c_vwn (7000): LDA: Vosko, Wilk, & Nussair lda_c_vwn_rpa (8000): LDA: Vosko, Wilk, & Nussair (fit to the RPA correlation energy) lda_c_pz (9000): LDA: Perdew & Zunger oep_x (901): OEP: Exact exchange
13 Articles published with Octopus Developer team Alberto Castro and Danilo Nitsche (Berlin) Miguel Marques (Lyon) Xavier Andrade and A. Rubio (San Sebastian) Micael Oliveira and Fernando Nogueira (Coimbra) Carlo Rozzi (Modena) Heiko Appel (San Diego) David Strubbe (Berkeley)
14 Example (ground state) run: Benzene molecule CalculationMode = gs Units = ev_angstrom radius = 5 spacing = 0.15 Output = wfs + density + elf + potential OutputHow = dx XYZCoordinates = "benzene.xyz" benzene.xyz: 12 C C C C C C H H H H H H Geometry of benzene (in Angstrom)
15 Useful ground state variable: Electron localization function (ELF) Motivation: How to visualize and how to give rigorous mathematical meanings to chemical bonds (single / double / triple)? Note: For this purpose, the density is not useful, nor the (Kohn Sham) orbitals.
16 Useful ingredients of ELF 1st order reduced density matrix: whose diagonal is the spin density diagonal of the 2nd order reduced density matrix: which gives the pair probability density probability of finding one electron at and another electron at
17 like spin conditional pair probability function probability of finding one spin electron at is another spin electron at, knowing that there coordinate transformation & spherical average probability of finding one spin electron at distance s from knowing that there is another spin electron at,
18 Taylor expansion measure of localization probability of finding a second like spin electron very near the reference electron if this probability is low then the reference electron must be very localized Define a useful visualization of localization Note: ELF is dimensionless and 3DEG = 3D uniform electron gas A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990)
19 Within DFT (or within any single determinantal wave function): A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990) ELF can be made time dependent within TDDFT (similar expression) T. Burnus, M. Marques, and E. K. U. Gross, PRA 71, (2005) ELF can be generalized to two dimensional systems E. R., A. Castro, and E. K. U. Gross, PRB 77, (2008)
20 ELF: Example A. Savin, R. Nesper, S. Wengert, and T. F. Fässler, Angew. Chem. Int. Ed. 36, 1808 (1997)
21 Time dependent ELF: Example TD ELF for the excitation of acetylene by a laser pulse. T. Burnus, M. Marques, and E. K. U. Gross, PRA 71, (2005)
22 Two dimensional ELF: Examples Density and ELF of 2D harmonic quantum dots Density and ELF of 2D harmonic four minima quantum dot molecules E. R., A. Castro, and E. K. U. Gross, PRB 77, (2008)
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