Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems.

Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new physical models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calculate response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact solution of the Schrödinger equation for low-dimensionality systems.

[1]  J. Werschnik,et al.  Optimal laser control of double quantum dots , 2007, 0707.0179.

[2]  University of Cambridge,et al.  THERMAL CONTRACTION AND DISORDERING OF THE AL(110) SURFACE , 1999 .

[3]  T. Beck Real-space mesh techniques in density-functional theory , 2000, cond-mat/0006239.

[4]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[5]  Xavier Andrade,et al.  Compressed Sensing for the Fast Computation of Matrices: Application to Molecular Vibrations , 2015, ACS central science.

[6]  E K U Gross,et al.  Controlling the dynamics of many-electron systems from first principles: a combination of optimal control and time-dependent density-functional theory. , 2010, Physical review letters.

[7]  Thomas Blumensath Fast compressed-sensing reconstruction for magnetic-resonance imaging , 2009 .

[8]  Tomoya Ono,et al.  Timesaving Double-Grid Method for Real-Space Electronic-Structure Calculations , 1999 .

[9]  Jan Mayer,et al.  A numerical evaluation of preprocessing and ILU-type preconditioners for the solution of unsymmetric sparse linear systems using iterative methods , 2009, TOMS.

[10]  P A Sterne,et al.  Real-space formulation of the electrostatic potential and total energy of solids , 2005 .

[11]  M. E. Galassi,et al.  GNU SCIENTI C LIBRARY REFERENCE MANUAL , 2005 .

[12]  David Taniar,et al.  Computational Science and Its Applications - ICCSA 2005, International Conference, Singapore, May 9-12, 2005, Proceedings, Part I , 2005, ICCSA.

[13]  Thomas Brabec,et al.  IONIZATION ABOVE THE COULOMB BARRIER , 1999 .

[14]  P. Wormer,et al.  Theory and Applications of Computational Chemistry The First Forty Years , 2005 .

[15]  Fernando Nogueira,et al.  Generating relativistic pseudo-potentials with explicit incorporation of semi-core states using APE, the Atomic Pseudo-potentials Engine , 2008, Comput. Phys. Commun..

[16]  Miguel A. L. Marques,et al.  Empirical functionals for reduced-density-matrix-functional theory , 2008, 0801.3332.

[17]  Stephan Kümmel,et al.  Simple iterative construction of the optimized effective potential for orbital functionals, including exact exchange. , 2003, Physical review letters.

[18]  Angel Rubio,et al.  Simulating pump-probe photoelectron and absorption spectroscopy on the attosecond timescale with time-dependent density functional theory. , 2013, Chemphyschem : a European journal of chemical physics and physical chemistry.

[19]  Herschel Rabitz,et al.  Monotonically convergent algorithm for quantum optimal control with dissipation , 1999 .

[20]  Emil Prodan,et al.  Quantum plasmonics: optical properties and tunability of metallic nanorods. , 2010, ACS nano.

[21]  Mark E. Casida,et al.  Time-dependent density-functional theory for molecules and molecular solids , 2009 .

[22]  Miguel A. L. Marques,et al.  Libxc: A library of exchange and correlation functionals for density functional theory , 2012, Comput. Phys. Commun..

[23]  Angel Rubio,et al.  Performance of nonlocal optics when applied to plasmonic nanostructures , 2013 .

[24]  Arai,et al.  Density-functional molecular dynamics with real-space finite difference. , 1995, Physical review. B, Condensed matter.

[25]  M. Petersilka,et al.  Excitation energies from time-dependent density-functional theory. , 1996 .

[26]  Miguel A L Marques,et al.  Benchmark calculations for reduced density-matrix functional theory. , 2008, The Journal of chemical physics.

[27]  R. Leeuwen,et al.  Exchange-correlation potential with correct asymptotic behavior. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[28]  J. C. Slater A Simplification of the Hartree-Fock Method , 1951 .

[29]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[30]  S G Louie,et al.  Coupling of nonlocal potentials to electromagnetic fields. , 2001, Physical review letters.

[31]  Angel Rubio,et al.  Ab initio nanoplasmonics: The impact of atomic structure , 2014 .

[32]  E. Rasanen,et al.  Femtosecond laser pulse shaping for enhanced ionization , 2009, 0906.1938.

[33]  J. Combes,et al.  Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions , 1971 .

[34]  Kresse,et al.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. , 1996, Physical review. B, Condensed matter.

[35]  Gregory Beylkin,et al.  Multiresolution quantum chemistry: basic theory and initial applications. , 2004, The Journal of chemical physics.

[36]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[37]  Paxton,et al.  High-precision sampling for Brillouin-zone integration in metals. , 1989, Physical review. B, Condensed matter.

[38]  Gernot Deinzer,et al.  Raman tensor calculated from the 2n+1 theorem in density-functional theory , 2002 .

[39]  A.M.K. Müller,et al.  Explicit approximate relation between reduced two- and one-particle density matrices , 1984 .

[40]  Gabriele Steidl,et al.  Fast Gauss transforms with complex parameters using NFFTs , 2006, J. Num. Math..

[41]  Rubio,et al.  Density-functional theory of the nonlinear optical susceptibility: Application to cubic semiconductors. , 1996, Physical review. B, Condensed matter.

[42]  G. Peano Sur une courbe, qui remplit toute une aire plane , 1890 .

[43]  Mario Piris,et al.  Iterative diagonalization for orbital optimization in natural orbital functional theory , 2009, J. Comput. Chem..

[44]  H. Rabitz,et al.  Control of quantum phenomena: past, present and future , 2009, 0912.5121.

[45]  Xavier Andrade,et al.  Time-dependent density functional theory scheme for efficient calculations of dynamic (hyper)polarizabilities. , 2007, The Journal of chemical physics.

[46]  J. Combes,et al.  A class of analytic perturbations for one-body Schrödinger Hamiltonians , 1971 .

[47]  Á. Rubio,et al.  Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theory , 2001, cond-mat/0102234.

[48]  George F. Bertsch,et al.  Time-dependent local-density approximation in real time , 1996 .

[49]  E. Gross,et al.  Fundamentals of time-dependent density functional theory , 2012 .

[50]  Xavier Andrade,et al.  Compressed Sensing for Multidimensional Spectroscopy Experiments. , 2012, The journal of physical chemistry letters.

[51]  A. J. Coleman THE STRUCTURE OF FERMION DENSITY MATRICES , 1963 .

[52]  Phillip F. Schewe Element 118 is discovered , 2006 .

[53]  Leigh S. Martin,et al.  Attosecond vacuum UV coherent control of molecular dynamics , 2013, Proceedings of the National Academy of Sciences.

[54]  Jean-Luc Fattebert,et al.  An inverse iteration method using multigrid for quantum chemistry , 1996 .

[55]  Liang Fu,et al.  Macroscopic polarization as a discrete Berry phase of the Hartree-Fock wave function: The single-point limit , 1998 .

[56]  Stephen Berkebile,et al.  Reconstruction of Molecular Orbital Densities from Photoemission Data , 2009, Science.

[57]  Á. Rubio,et al.  octopus: a first-principles tool for excited electron-ion dynamics. , 2003 .

[58]  Xavier Andrade,et al.  Real-Space Density Functional Theory on Graphical Processing Units: Computational Approach and Comparison to Gaussian Basis Set Methods. , 2013, Journal of chemical theory and computation.

[59]  Xavier Andrade,et al.  Modified Ehrenfest Formalism for Efficient Large-Scale ab initio Molecular Dynamics. , 2008, Journal of chemical theory and computation.

[60]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[61]  L. Reining,et al.  Electronic excitations: density-functional versus many-body Green's-function approaches , 2002 .

[62]  Sullivan,et al.  Real-space multigrid-based approach to large-scale electronic structure calculations. , 1996, Physical review. B, Condensed matter.

[63]  C. Sauer,et al.  Complete determination of molecular orbitals by measurement of phase symmetry and electron density , 2014, Nature Communications.

[64]  W. Kutzelnigg,et al.  Theory of magnetic susceptibilities and NMR chemical shifts in terms of localized quantities. II. Application to some simple molecules , 1982 .

[65]  H. Rabitz,et al.  Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications. , 1988, Physical review. A, General physics.

[66]  N. Mermin Thermal Properties of the Inhomogeneous Electron Gas , 1965 .

[67]  Tobias Burnus,et al.  Time-dependent electron localization function , 2005 .

[68]  T. Gilbert Hohenberg--Kohn theorem for nonlocal external potentials , 1975 .

[69]  Kikuji Hirose,et al.  Total-energy minimization of few-body electron systems in the real-space finite-difference scheme , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[70]  Peter Pulay,et al.  Efficient implementation of the gauge-independent atomic orbital method for NMR chemical shift calculations , 1990 .

[71]  David A. Strubbe Optical and Transport Properties of Organic Molecules: Methods and Applications , 2012 .

[72]  R. C. Haddon,et al.  Magnetism of the carbon allotropes , 1995, Nature.

[73]  A. Siegert On the Derivation of the Dispersion Formula for Nuclear Reactions , 1939 .

[74]  N. Hatano,et al.  Some Properties of the Resonant State in Quantum Mechanics and Its Computation , 2007, 0705.1388.

[75]  Kikuji Hirose,et al.  Real-space finite-difference approach for multi-body systems: path-integral renormalization group method and direct energy minimization method , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

[76]  Yew Kam Ho,et al.  The method of complex coordinate rotation and its applications to atomic collision processes , 1983 .

[77]  Xavier Andrade,et al.  Optical and magnetic properties of boron fullerenes. , 2009, Physical chemistry chemical physics : PCCP.

[78]  Samuel H. Tersigni,et al.  Wavepacket dancing: Achieving chemical selectivity by shaping light pulses , 1989 .

[79]  Jean-François Méhaut,et al.  High Performance Computing / Le Calcul Intensif Daubechies wavelets for high performance electronic structure calculations: The BigDFT project , 2011 .

[80]  A. Szabó,et al.  Modern quantum chemistry : introduction to advanced electronic structure theory , 1982 .

[81]  J. Werschnik,et al.  Quantum optimal control theory , 2007, 0707.1883.

[82]  Kwang S. Kim,et al.  Theory and applications of computational chemistry : the first forty years , 2005 .

[83]  X. Andrade,et al.  Efficient formalism for large-scale ab initio molecular dynamics based on time-dependent density functional theory. , 2007, Physical review letters.

[84]  Yosuke Kanai,et al.  Quantum Dynamics Simulation of Electrons in Materials on High-Performance Computers , 2014, Computing in Science & Engineering.

[85]  Pekka Koskinen,et al.  Structural relaxation made simple. , 2006, Physical review letters.

[86]  Jeremy J. Baumberg,et al.  Revealing the quantum regime in tunnelling plasmonics , 2012, Nature.

[87]  Galli,et al.  Real-space adaptive-coordinate electronic-structure calculations. , 1995, Physical review. B, Condensed matter.

[88]  Tomoya Ono,et al.  Real-space calculations for electron transport properties of nanostructures , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

[89]  Stefan Kunis,et al.  Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms , 2009, TOMS.

[90]  Miguel A. L. Marques,et al.  Optimal control of the electronic current density: Application to one- and two-dimensional one-electron systems , 2011, 1101.1429.

[91]  U. Kaldor,et al.  N2 excitations below 15 eV by the multireference coupled‐cluster method , 1990 .

[92]  William P. Reinhardt,et al.  Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics , 1982 .

[93]  Angel Rubio,et al.  Modeling electron dynamics coupled to continuum states in finite volumes with absorbing boundaries , 2014 .

[94]  J D Ramsden,et al.  Exact density-functional potentials for time-dependent quasiparticles. , 2012, Physical review letters.

[95]  Angel Rubio,et al.  Modelling the effect of nuclear motion on the attosecond time-resolved photoelectron spectra of ethylene , 2014, 1403.5408.

[96]  Garnett W. Bryant,et al.  Plasmonic properties of metallic nanoparticles: The effects of size quantization , 2010, CLEO: 2011 - Laser Science to Photonic Applications.

[97]  R. Orlando,et al.  CRYSTAL14: A program for the ab initio investigation of crystalline solids , 2014 .

[98]  Adam Wasserman,et al.  Hohenberg-Kohn theorem for the lowest-energy resonance of unbound systems. , 2007, Physical review letters.

[99]  Rubio,et al.  Dielectric screening effects on the photoabsorption cross section of embedded metallic clusters. , 1993, Physical review. B, Condensed matter.

[100]  E. Gross,et al.  Optimal control theory for quantum-classical systems: Ehrenfest molecular dynamics based on time-dependent density-functional theory , 2013, 1308.4162.

[101]  Jürg Hutter,et al.  Excited state nuclear forces from the Tamm–Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework , 2003 .

[102]  S. H. Vosko,et al.  Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis , 1980 .

[103]  Javier Aizpurua,et al.  Bridging quantum and classical plasmonics with a quantum-corrected model , 2012, Nature Communications.

[104]  Á. Rubio,et al.  J un 2 00 1 Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in TDDFT , 2001 .

[105]  Eduardo Hernández,et al.  A new real-space algorithm for realistic density functional calculations , 2007 .

[106]  Aage E. Hansen,et al.  Ab initio calculations of oscillator and rotatory strengths in the random‐phase approximation: Twisted mono‐olefins , 1977 .

[107]  B. M. Fulk MATH , 1992 .

[108]  Yan Li,et al.  Exact relations in the optimized effective potential method employing an arbitrary Exc[}ψiσ{] , 1990 .

[109]  John P. Perdew,et al.  A self-interaction corrected approach to many-electron systems: Beyond the local spin density approximation , 1980 .

[110]  Manfred Lein,et al.  Strong-field ionization dynamics of a model H 2 molecule , 2002 .

[111]  James R. Chelikowsky,et al.  Excited-state forces within time-dependent density-functional theory: A frequency-domain approach , 2007 .

[112]  Adam Wasserman,et al.  Resonance Lifetimes from Complex Densities , 2009, 0910.4599.

[113]  A. D. Boardman,et al.  Electromagnetic surface modes , 1982 .

[114]  Xavier Andrade,et al.  Measurement of the absolute Raman cross section of the optical phonons in type Ia natural diamond , 2012 .

[115]  Mario Piris,et al.  A Natural Orbital Functional Based on an Explicit Approach of the Two-Electron Cumulant , 2013 .

[116]  Osamu Sugino,et al.  Analytical expression for the excited-state force from density-functional perturbation theory , 2012 .

[117]  Benjamin T. Miller,et al.  A parallel implementation of the analytic nuclear gradient for time-dependent density functional theory within the Tamm–Dancoff approximation , 1999 .

[118]  David Taniar,et al.  Computational Science and Its Applications – ICCSA 2014 , 2014, Lecture Notes in Computer Science.

[119]  Evert Jan Baerends,et al.  An improved density matrix functional by physically motivated repulsive corrections. , 2005, The Journal of chemical physics.

[120]  N. A. Romero,et al.  Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[121]  D. Chong Recent Advances in Density Functional Methods Part III , 2002 .

[122]  Marco Buongiorno Nardelli,et al.  O(N) real-space method for ab initio quantum transport calculations: Application to carbon nanotube-metal contacts , 2001 .

[123]  Stefan A. Maier,et al.  Quantum Plasmonics , 2016, Proceedings of the IEEE.

[124]  A. Becke Basis‐set‐free density‐functional quantum chemistry , 2009 .

[125]  Francesco Mauri,et al.  Nonlocal pseudopotentials and magnetic fields. , 2003, Physical review letters.

[126]  Allen Taflove,et al.  Application of the Finite-Difference Time-Domain Method to Sinusoidal Steady-State Electromagnetic-Penetration Problems , 1980, IEEE Transactions on Electromagnetic Compatibility.

[127]  A. Jauho,et al.  Unusual resonances in nanoplasmonic structures due to nonlocal response , 2011, 1106.2175.

[128]  Barry Simon,et al.  Resonances in n-Body Quantum Systems With Dilatation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory , 1973 .

[129]  Xavier Andrade,et al.  Towards a gauge invariant method for molecular chiroptical properties in TDDFT. , 2009, Physical chemistry chemical physics : PCCP.

[130]  A. Borisov,et al.  Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response. , 2013, Physical review letters.

[131]  H. Appel,et al.  Time-dependent density-functional and reduced density-matrix methods for few electrons: Exact versus adiabatic approximations , 2011, 1108.3196.

[132]  Angel Rubio,et al.  Stark Ionization of Atoms and Molecules within Density Functional Resonance Theory , 2013, 1309.2925.

[133]  Adam Wasserman,et al.  Density functional resonance theory of unbound electronic systems. , 2011, Physical review letters.

[134]  Mark Earl Casida,et al.  In Recent Advances in Density-Functional Methods , 1995 .

[135]  Malcolm J. Stott,et al.  Calculation of vibrational frequencies within the real space pseudopotential approach , 2005 .

[136]  A. Nakano,et al.  Divide-and-conquer density functional theory on hierarchical real-space grids: Parallel implementation and applications , 2008 .

[137]  Dennis R. Salahub,et al.  Dynamic polarizabilities and excitation spectra from a molecular implementation of time‐dependent density‐functional response theory: N2 as a case study , 1996 .

[138]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[139]  E. Gross,et al.  Optimization schemes for selective molecular cleavage with tailored ultrashort laser pulses , 2011, 1102.3128.

[140]  Mykhaylo Krykunov,et al.  On the relation between time-dependent and variational density functional theory approaches for the determination of excitation energies and transition moments. , 2009, The Journal of chemical physics.

[141]  S deGironcoli,et al.  Lattice dynamics of metals from density-functional perturbation theory. , 1995 .

[142]  Xavier Andrade Linear and non-linear response phenomena of molecular systems within time-dependent density functional theory , 2010 .

[143]  Angel Rubio,et al.  Real-space, real-time method for the dielectric function , 2000 .

[144]  P. Decleva,et al.  Time-dependent density-functional theory for molecular photoionization with noniterative algorithm and multicenter B-spline basis set: CS2 and C6H6 case studies. , 2005, The Journal of chemical physics.

[145]  John P. Perdew,et al.  Jacob’s ladder of density functional approximations for the exchange-correlation energy , 2001 .

[146]  John Skilling Programming the Hilbert curve , 2004 .

[147]  P. Elliott,et al.  Absence of dynamical steps in the exact correlation potential in the linear response regime , 2013 .

[148]  A. Wasserman,et al.  Derivative discontinuities in density functional theory , 2014 .

[149]  Fred H. Pollak,et al.  Energy-Band Structure of Germanium and Silicon: The k [] p Method , 1966 .

[150]  H. G. Muller,et al.  Time-Resolved Holography with Photoelectrons , 2011, Science.

[151]  Marco Milla,et al.  Radar imaging with compressed sensing , 2013 .

[152]  M. Casula,et al.  Density functional theory beyond the linear regime: Validating an adiabatic local density approximation , 2011, 1101.2564.

[153]  Angel Rubio,et al.  Time-dependent density functional approach for the calculation of inelastic x-ray scattering spectra of molecules. , 2010, The Journal of chemical physics.

[154]  Herschel Rabitz,et al.  A RAPID MONOTONICALLY CONVERGENT ITERATION ALGORITHM FOR QUANTUM OPTIMAL CONTROL OVER THE EXPECTATION VALUE OF A POSITIVE DEFINITE OPERATOR , 1998 .

[155]  Lei Zhu,et al.  Faster STORM using compressed sensing , 2012, Nature Methods.

[156]  Daan Frenkel,et al.  Visualizing basins of attraction for different minimization algorithms. , 2013, The journal of physical chemistry. B.

[157]  A. Castro,et al.  Theoretical shaping of femtosecond laser pulses for ultrafast molecular photo-dissociation with control techniques based on time-dependent density functional theory. , 2013, Chemphyschem : a European journal of chemical physics and physical chemistry.

[158]  E K U Gross,et al.  Optimal control of quantum rings by terahertz laser pulses. , 2007, Physical review letters.

[159]  H. Rabitz,et al.  RAPIDLY CONVERGENT ITERATION METHODS FOR QUANTUM OPTIMAL CONTROL OF POPULATION , 1998 .

[160]  Micael J. T. Oliveira,et al.  Toward an All-Around Semilocal Potential for Electronic Exchange , 2010 .

[161]  Angel Rubio,et al.  Propagators for the time-dependent Kohn-Sham equations. , 2004, The Journal of chemical physics.

[162]  広瀬 喜久治,et al.  First-principles calculations in real-space formalism : electronic configurations and transport properties of nanostructures , 2005 .

[163]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[164]  M. Tsukada,et al.  Electronic-structure calculations based on the finite-element method. , 1995, Physical review. B, Condensed matter.

[165]  R. Nieminen,et al.  Real-space electronic-structure calculations: Combination of the finite-difference and conjugate-gradient methods. , 1995, Physical review. B, Condensed matter.

[166]  Rochus Schmid,et al.  A general and efficient pseudopotential Fourier filtering scheme for real space methods using mask functions. , 2006, The Journal of chemical physics.

[167]  X. Gonze,et al.  Density-functional approach to nonlinear-response coefficients of solids. , 1989, Physical review. B, Condensed matter.

[168]  Stefano de Gironcoli,et al.  QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[169]  Xavier Andrade,et al.  Cluster-surface and cluster-cluster interactions: Ab initio calculations and modeling of asymptotic van der Waals forces , 2008, 0806.2946.

[171]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[172]  Michael P. Friedlander,et al.  Probing the Pareto Frontier for Basis Pursuit Solutions , 2008, SIAM J. Sci. Comput..

[173]  Naomi J. Halas,et al.  Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles , 2004 .

[174]  Xavier Andrade,et al.  Prediction of the derivative discontinuity in density functional theory from an electrostatic description of the exchange and correlation potential. , 2011, Physical review letters.

[175]  Rochus Schmid,et al.  Car–Parrinello molecular dynamics using real space wavefunctions , 2006 .

[176]  J. Bernholc,et al.  Recent developments and applications of the real-space multigrid method , 2008 .

[177]  Taisuke Boku,et al.  A massively-parallel electronic-structure calculations based on real-space density functional theory , 2010, J. Comput. Phys..

[178]  David A. Strubbe,et al.  BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures , 2011, Comput. Phys. Commun..

[179]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[180]  Simpson,et al.  Static, dynamic, and electronic properties of liquid gallium studied by first-principles simulation. , 1995, Physical review. B, Condensed matter.

[181]  Weitao Yang,et al.  Insights into Current Limitations of Density Functional Theory , 2008, Science.

[182]  A. Borisov,et al.  Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer. , 2012, Nano letters.

[183]  H. Rabitz,et al.  Optimal control of selective vibrational excitation in harmonic linear chain molecules , 1988 .

[184]  F. A. Thiel,et al.  Experimental and theoretical determination of the magnetic susceptibility of C60 and C70 , 1991, Nature.

[185]  Y. Saad,et al.  Finite-difference-pseudopotential method: Electronic structure calculations without a basis. , 1994, Physical review letters.

[186]  Xavier Andrade,et al.  Basis set effects on the hyperpolarizability of CHCl3: Gaussian-type orbitals, numerical basis sets and real-space grids. , 2010, The Journal of chemical physics.

[187]  Stefano de Gironcoli,et al.  Phonons and related crystal properties from density-functional perturbation theory , 2000, cond-mat/0012092.

[188]  Xavier Andrade,et al.  A survey of the parallel performance and accuracy of Poisson solvers for electronic structure calculations , 2012, J. Comput. Chem..

[189]  Xavier Gonze,et al.  Dynamical matrices, born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory , 1997 .

[190]  John Francis Dobson,et al.  High-frequency hydrodynamics and Thomas–Fermi theory ☆ , 2000 .

[191]  A. Rubio,et al.  Ab-initio angle and energy resolved photoelectron spectroscopy with time-dependent density-functional theory , 2012 .

[192]  Y. Saad,et al.  PARSEC – the pseudopotential algorithm for real‐space electronic structure calculations: recent advances and novel applications to nano‐structures , 2006 .

[193]  X. Andrade,et al.  Application of compressed sensing to the simulation of atomic systems , 2012, Proceedings of the National Academy of Sciences.

[194]  R. Mcweeny,et al.  Methods Of Molecular Quantum Mechanics , 1969 .

[195]  D. Hilbert,et al.  Ueber die reellen Züge algebraischer Curven , 1891 .

[196]  Á. Rubio,et al.  Time-dependent density-functional theory. , 2009, Physical chemistry chemical physics : PCCP.

[197]  Xu Zhang,et al.  Size-Dependent Plasmonic Resonances from Large-Scale Quantum Simulations. , 2014, The journal of physical chemistry letters.

[198]  White,et al.  Finite-element method for electronic structure. , 1989, Physical review. B, Condensed matter.

[199]  Thomas L. Beck,et al.  Real‐space multigrid solution of electrostatics problems and the Kohn–Sham equations , 1997 .