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Before we start: Important setup of your Computer change directory: cd /afs/ictp/public/shared/smr2475./setup-config.sh logout login again

1 st Tutorial: The Basics of DFT Lydia Nemec and Oliver T. Hofmann Fritz Haber Institute of the Max Planck Society 7th August 2013 Density functional theory and beyond: Computational materials science for real materials, 2013

The ultimate goal! HΨ = EΨ Second order differential equation for a 3N e -variable function Ψ Complex problem Unsolved issues at the simplest level of approximations (mulitiple solutions,...)

Goals of this tutorial Familiarise with practical aspects of electronic structure theory in general and density functional theory (DFT) in particular Hartree-Fock (HF) method and Kohn-Sham DFT (non-periodic) Numerical solution of the approximate equations (tool: FHI-aims) Exploring potential energy surfaces (total energies at fixed nuclei, local minima, vibrational spectra) Electronic structure analysis (visualisation tools, electron density, Kohn-Sham orbitals and spectrum)

Solving the Kohn-Sham equations Hohenberg-Kohn Theorem Ψ(r 1...r Ne ) n(r) Kohn-Sham scheme ( 1 2 2 + d 3 r n(r ) ) r r + v xc + v ψ i = ɛ i ψ i n = i f i ψ i 2 KS Orbitals {ψ i } ψ i, ψ j = δ ij XC Potential v xc unknown,but many approximations exist LDA, PBE,... control.in External potential contains ionic contributions geometry.in

Basis set N Expand in a finite basis {φ i }: ψ j = c ij φ i i=1 Finite Basis Numeric atom centered Gaussians Plane waves + pseudoisation Slater type Grid based... many more

Basis set N Expand in a finite basis {φ i }: ψ j = c ij φ i i=1 Numeric atom centered (FHI-aims) φ i (r) = u i(r) Y lm (Ω) r Flexible: Free-atom like Hydrogen like Free Ions and many more... Courtesy V. Blum

Basis set N Expand in a finite basis {φ i }: ψ j = c ij φ i i=1 Generalized matrix eigenvalue equation in c ij ĥ KS ψ = Eψ j h ij (c)c jl = ɛ l s ij c jl Overlap matrix s ij = φ i, φ j Hamilton matrix h ij = φ i, ĥ KS φ j Self-consistent solution j

Finding the self-consistent solution i = 0; Initialize i++ Update density Solve EV equation Mixing (Pulay) (Sca)Lapack ELPA no i > 0 converged? yes Have a nice day! Criteria Energy Charge density Sum of eigenvalues Force

Problem I: The hydrogen atom Tasks: Input files needed to run FHI-aims. Test the convergence of the total energy with basis size. Compare the total energy of the hydrogen atom computed with different methods implemented in FHI-aims. Do all methods converge to the same result?

FHI-aims: input files geometry.in control.in

FHI-aims: input files geometry.in # Atomic structure # x y z atom 0.0 0.0 0.0 H atom 1.0 0.0 0.0 H initial_moment 1.0 control.in # That s a comment Units: Positions in Å Energies in ev Manual, chap. 2.1

FHI-aims: input files geometry.in # Atomic structure # x y z atom 0.0 0.0 0.0 H atom 1.0 0.0 0.0 H initial_moment 1.0 control.in # Physical model settings xc pw-lda charge 0. spin collinear # That s a comment Units: Positions in Å Energies in ev Manual, chap. 2.1

FHI-aims: input files geometry.in # Atomic structure # x y z atom 0.0 0.0 0.0 H atom 1.0 0.0 0.0 H initial_moment 1.0 # That s a comment Units: Positions in Å Energies in ev control.in # Physical model settings xc pw-lda charge 0. spin collinear # SCF convergence settings sc_accuracy_eev 1E-2 sc_accuracy_etot 1E-5 sc_accuracy_rho 1E-4 sc_iter_limit 100 Manual, chap. 2.1

FHI-aims: input files geometry.in # Atomic structure # x y z atom 0.0 0.0 0.0 H atom 1.0 0.0 0.0 H initial_moment 1.0 # That s a comment Units: Positions in Å Energies in ev control.in # Physical model settings xc pw-lda charge 0. spin collinear # SCF convergence settings sc_accuracy_eev 1E-2 sc_accuracy_etot 1E-5 sc_accuracy_rho 1E-4 sc_iter_limit 100 # Species specifics... Manual, chap. 2.1

species_default /usr/local/aimsfiles/species_default Predefined species Manual, chap. 2.2 Copy-paste into control.in light tight really tight

species_default /usr/local/aimsfiles/species_default Predefined species Manual, chap. 2.2 Copy-paste into control.in light tight really tight Increased accuracy: Basis Hartree potential Basis cutoff potential Integration grids

species_default /usr/local/aimsfiles/species_default Predefined species Manual, chap. 2.2 Copy-paste into control.in light tight really tight Fast, many production tasks Fast pre-relaxation Used to verify important results Converged settings Heavily converged numerical settings Explicit convergence tests

species_default /usr/local/aimsfiles/species_default Predefined species Manual, chap. 2.2 Copy-paste into control.in light tight really tight Fast, many production tasks Fast pre-relaxation Used to verify important results Converged settings Heavily converged numerical settings Explicit convergence tests Additionally converge basis ("tiers")!

FHI-aims output 1 Invoking FHI-aims... Introduction

FHI-aims output 1 2 Invoking FHI-aims... Reading file control.in. - Summary of control.in file

FHI-aims output 1 2 3 Invoking FHI-aims... Reading file control.in. - - Reading geometry description geometry.in. Summary of geometry.in file

FHI-aims output 1 2 3 4 Invoking FHI-aims... Reading file control.in. - - Reading geometry description geometry.in. Preparing all fixed parts of the calculation. Geometry independent preparations Basis set generation

FHI-aims output 5 - Begin self-consistency loop: Initialization. Date : 20130610, Time : 162002.389 Geometry dependent preparations Integration grid Initialization of charge density

FHI-aims output 5 6 - Begin self-consistency loop: Initialization. Date : 20130610, Time : 162002.389 - Begin self-consistency iteration # 1 Date : 20130610, Time : 162002.445 First SCF cycle

FHI-aims output 5 6 THIS TUTORIAL - Begin self-consistency loop: Initialization. Date : 20130610, Time : 162002.389 - Begin self-consistency iteration # 1 Date : 20130610, Time : 162002.445 First SCF cycle Periodic metals only» Energy Total energy : -13.01991124 ev Total energy, T -> 0 : -13.01991124 ev Electronic free energy : -13.01991124 ev

FHI-aims output 5 6 - Begin self-consistency loop: Initialization. Date : 20130610, Time : 162002.389 - Begin self-consistency iteration # 1 Date : 20130610, Time : 162002.445 First SCF cycle» Self-consistency convergence accuracy Change of charge density : 0.6753E-02 Change of sum of eigenvalues : 0.4376E+00 ev Change of total energy : 0.1143E-01 ev

FHI-aims output 5 6 - Begin self-consistency loop: Initialization. Date : 20130610, Time : 162002.389 - Begin self-consistency iteration # 6 Date : 20130610, Time : 162002.560 Sixth SCF cycle» Self-consistency convergence accuracy Change of charge density : 0.3163E-05 Change of sum of eigenvalues : -.9415E-05 ev Change of total energy : 0.2388E-10 ev

FHI-aims output 7 Self-consistency cycle converged.

FHI-aims output 7 Self-consistency cycle converged.» Energy and forces Total energy uncorrected : -0.130198526094581E+02 ev Total energy corrected : -0.130198526094581E+02 ev Electronic free energy : -0.130198526094581E+02 ev» SCF info Number of self-consistency cycles : 6» Timings

FHI-aims output 7 Self-consistency cycle converged.» Energy and forces Total energy uncorrected : -0.130198526094581E+02 ev Total energy corrected : -0.130198526094581E+02 ev Electronic free energy : -0.130198526094581E+02 ev» SCF info Number of self-consistency cycles : 6» Timings 8 Have a nice day.

FHI-aims output 7 Self-consistency cycle converged. Postprocessing Structure optimization» Get next relaxation step» Redo SCF for new geometry 8 Have a nice day.

Problem II: Hydrofluoric acid (HF) One of the first papers which systematically investigated the performance of DFT was published by John A. Pople and coworker in 1993. Tasks: Find the equilibrium bond distance of HF. Compare the HF bond length for different methods. Calculate the atomization energy ( H at ). Compute the dipole moment for different methods and bond lengths. B. G. Johnson, P. M. W. Gill and J. A. Pople, The performance of a family of density functional methods, The Journal of Chemical Physics 98, 5612 (1993).

Forces E X

Forces Energy gradient E X

Forces Structure optimization: Find local minimum on potential energy surface (PES) E? X

Forces Structure optimization: Find local minimum on potential energy surface (PES) E? Many methods! Industry standard: quasi Newton methods X

Forces Structure optimization: Find local minimum on potential energy surface (PES) E? # Specify in FHI-aims sc_accuracy_forces 1E-4 relax_geometry trm 1E-3 X

Local structure optimisation Problem IV to VIII: Hydronium cation (H 3 O + ) Tasks: Relax structure with two different starting points. Make a vibrational analysis. Explore the limits of the harmonic approximation.

Local structure optimisation Harmonic molecular motion How do atoms move in a potential V? Solve equations of motion! Kinetic energy: T Potential energy: V d T + V = 0 dt Ṙ i R i

Local structure optimisation Harmonic molecular motion How do atoms move in a potential V? Solve equations of motion! Kinetic energy: T Potential energy: V d T + V = 0 dt Ṙ i R i Taylor expansion of V around equilibrium position R 0 + harmonic approximation F : Forces H: Hessian V = V 0 F (R 0 )R + 1 }{{} 2 RT H(R 0 )R +...higher terms }{{} =0 equilibrium =0 harmonic approximation

Local structure optimisation Harmonic molecular motion How do atoms move in a potential V? Solve equations of motion! Kinetic energy: T Potential energy: V d T dt Ṙi Solution The Dynamic Matrix D ij : D ij = + V R i = 0 1 Mi Mj H ij Eigenmodes Q If R Qe iωt, with DQ ω 2 Q = 0 (1) Harmonic approximation is valid (2) Equilibrium geometry

Local structure optimisation Vibrations Solve R 3N R 3N eigenvalue equation det(d ω 2 1) = 0 Hessian H Dynamic matrix D H ij := 2 E 1 R i R j D ij = H ij Mi Mj In practice: finite central numerical differences (of forces) Wrapper > aims_vibrations.mpi.pl Manual, chap 4.6

Local structure optimisation Vibrations Solve R 3N R 3N eigenvalue equation det(d ω 2 1) = 0 Get» Eigenmodes {Q i, i 1... 3N}» Eigenfrequencies

Local structure optimisation Vibrations Solve R 3N R 3N eigenvalue equation det(d ω 2 1) = 0 Get» Eigenmodes {Q i, i 1... 3N}» Eigenfrequencies» 6 (almost) zero frequency modes (if molecule non-linear) translations + rotations» Imaginary frequency Saddle point

Local structure optimisation Vibrations Solve R 3N R 3N eigenvalue equation det(d ω 2 1) = 0 Get» Eigenmodes {Q i, i 1... 3N}» Eigenfrequencies» 6 (almost) zero frequency modes (if molecule non-linear) translations + rotations» Imaginary frequency Saddle point» Infrared intensities (derivative of dipole moment µ) I i dµ dq i 2

Local structure optimisation Vibrations Get Based on harmonic approximation! Limitation: Problem IX Beyond: Tutorial 3 (MD) on Friday» Eigenmodes {Q i, i 1... 3N}» Eigenfrequencies» 6 (almost) zero frequency modes (if molecule non-linear) translations + rotations» Imaginary frequency Saddle point» Infrared intensities (derivative of dipole moment µ) I i dµ 2 dq i

Visualising density differences Visualization Orbitals and densities Keyword in control.in output cube eigenstate homo cube filename HOMO.cube output cube total_density cube filename tot_dens_uc.cube Get: *.cube file - values on a regular 3D grid. Software: molden (jmol, gdis, xcrysden) Appendix of handout

Practical issues Each calculation one directory > mkdir tutorial1 > cd tutorial1 > mkdir HF 2 input files geometry.in control.in Launching FHI-aims calculation mpirun -np 4 aims.hands-on-2013.scalapack.mpi.x tee aims.out... scripting helps! (Sample scripts in appendix of handout)

Before we start: Important setup of your Computer change directory: cd /afs/ictp/public/shared/smr2475./setup-config.sh logout from your KDE session login again Access the info-lab machine from outside via ssh: Access the ICTP gateway ssh your_user_name@ssh.ictp.it From the gateway you can reach your workstation: ssh hp83-inf-xx replace XX by the number of your maschine (XX=1,..51)

Acknowledgements Special Thanks! Viktor Atalla Adriana Supady

Introduction Part I Part II Part III Your Tutors for the afternoon Oliver T. Hofmann Lydia Nemec Christian Carbogno Franz Knuth Arvid Ihrig Markus Schneider Volker Blum

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