Plane waves and pseudopotentials (Part II)

Transcription

1 Bristol 2007 Plane waves and pseudopotentials (Part II) X. Gonze Université Catholique de Louvain, Louvain-la-neuve, Belgium Plane Waves - Pseudopotentials, Bristol March

2 In view of practical calculations, what did we learn? a 3 We have to specify... 1) A starting geometry : - definition of a cell - list and position of atoms - pseudopotentials 2) A band occupation scheme (+number of bands to be computed) 3) A grid for sampling of the BZ 4) Of course, an approximation to DFT 5) A plane wave basis set (sphere, through kinetic cut-off energy) We will have some FFT real space grid, automatically determined, from the planewave sphere, with doubled radius... V PS (R) (Ha) Plane Waves - Pseudopotentials, Bristol March a 2 a 1! Z v r Radial distance (atomic units) d N PW (2 E cut ) 1 2

3 Outline I. Formalism II. III. (the PW basis set, Brillouin Zone integration, pseudopotentials, computing the forces, the PAW method) Iterative techniques (the Kohn-Sham equation, the SCF convergence, optimization of the geometry) Applications (the growth of Carbon nanotubes, the index of refraction of crystal ) IV. The ABINIT software (licence, structure, capabilities) Plane Waves - Pseudopotentials, Bristol March

4 Applications : growth of carbon nanotubes

5 Growth of Carbon nanotubes Stability of monolayer nanotubes at high temperature (10,0) T=3000K T=0K T=1500K Spontaneous closing as a graphitic dome : no dangling bond at the tube termination. No further inclusion of carbon atoms. Charlier, De Vita, Blase, Car, Science 275, 646 (1997) Plane Waves - Pseudopotentials, Bristol March

6 Growth of Carbon nanotubes : the influence of a catalyst One atom of Nickel or Cobalt (in red) allows the growth to proceed. T=0K 2000K T=0K Spontaneous closing of the CNT, even in presence of a catalyst. However, incoming carbon atoms are readily incorporated... Charlier, et al., unpublished Plane Waves - Pseudopotentials, Bristol March

7 Transition state analysis Small energy barrier for incorporation in the graphitic network Moreover, the reconstruction of the graphitic termination is facilitated ( Stone-Wales mechanism) Plane Waves - Pseudopotentials, Bristol March

8 Application : refraction index of crystal

9 The effect of lead in SiO 2 So-called Crystal glassware : the presence of lead increases the refraction index of silica glass. How is the index of refraction changed? Where are the lead atoms located in the SiO 2 framework? One needs to be able to treat accurately the electronic ground state, as well as the response of the electrons to light. Plane Waves - Pseudopotentials, Bristol March

10 Two important mechanisms : substitution, creation of vacancies (Red : oxygen atoms ; blue : silicon atoms ; yellow : lead atom) Pb Si ΔE f GGA = 0.16 ev with µ Si Min and µ Pb metal Pb Si + V O ΔE f GGA = 1.72 ev with µ Si Min and µ Pb metal Plane Waves - Pseudopotentials, Bristol March

11 Definition of the formation energy E f [Pb i ]=E tot [SiO 2.Pb]-N Si µ Si -N O2 µ O2 -N Pb µ Pb -N e -ε F N si, N O2, N Pb : # atoms of Si, Pb and molecules of O 2 µ Si, µ O 2, µ Pb : chemical potentials of Si, O 2 et Pb N e - : # e- transferred from a chemical bath at ε F Silicon and oxygen in equilibrium with quartz : µ Si +µ O2 = µ SiO2 Plane Waves - Pseudopotentials, Bristol March

12 Map of allowed chemical potentials Lead rich Silicon Poor /Oxygen rich Silicon rich Plane Waves - Pseudopotentials, Bristol March

13 Phases that define the bounds on chemical potential Pure silicon Pure lead Gaseous oxygen Alamosite PbSiO 3 + PbO 2 -β PbO-α Pb 3 O 4 Plane Waves - Pseudopotentials, Bristol March

14 Some technical information Representation of the atoms in a supercell of 72 atoms (384 valence electrons) Representation of the electronic wavefunctions by plane waves, up to a kinetic energy of Hartree (on the order of plane waves) Computation of forces on each atom, and stresses on the cell, for fixed geometric configuration, then optimization of the geometry following the forces and stresses, such as to determine the atomic bond lengths and cell parameters to better than 0.2 % => Determination of the VOLUME OF FORMATION, for the different models of lead in silica, with comparison to crystalline quartz => Determination of the total energy of the models ENERGY OF => Determination of the total energy of sinks FORMATION => Determination of the response of the model to an applied homogeneous electric field Plane Waves - Pseudopotentials, Bristol March

15 Other configurations of lead in silica Pb O Pb I (Interstitial) Pb Si 2 O Not an exhaustive search! Plane Waves - Pseudopotentials, Bristol March

16 Results Volume of formation (cubic Bohr cubic Angstrom) PbSi PbSi + VO PbI PbO PbSi2O Energy of formation (ev) Sinks : lead rich (metallic lead), oxygen rich (minimal chemical potential of Si) Plane Waves - Pseudopotentials, Bristol March

17 Modification of the refractive index due to lead Refractive index Comparison : quartz+pb vs flint Different pure silica value, but slope in excellent agreement Limited influence of the atomic geometry on the effect of lead Increase of the refractive index due to the intrinsic polarisability of lead atoms (Detraux and Gonze, unpublished) Plane Waves - Pseudopotentials, Bristol March

18 Outline I. Formalism II. III. (the PW basis set, Brillouin Zone integration, pseudopotentials, computing the forces, the PAW method) Iterative techniques (the Kohn-Sham equation, the SCF convergence, optimization of the geometry) Applications (the growth of Carbon nanotubes, the index of refraction of crystal ) IV. The ABINIT software (licence, structure, capabilities) Plane Waves - Pseudopotentials, Bristol March

19 The Free or Open Source software concept Free for freedom, not price freedom 1 : unlimited use for any purpose freedom 2 : study and modify for your needs (need source access!) freedom 3 : copy freedom 4 : distribute modifications From copyright to freedom ( copyleft ) copyright allows licensing licenses grants freedom Here : the GNU General Public License (GPL), like Linux. Web site : http: // Plane Waves - Pseudopotentials, Bristol March

20 ABINIT v5.3 capabilities (I) Methodologies Pseudopotentials/Plane Waves + Projector Augmented Waves (for selected capabilities) Many pseudopotential types, different PAW generators + Wavelets (stage where SCF and forces are OK, to be followed by order(n) implementation) Density functionals : LDA, GGA (PBE and variations, HCTH), LDA+U (or GGA+U) + some advanced functionals (exact exchange + RPA or...) TDDFT for finite systems excitation energies (Casida) GW for accurate electronic eigenenergies (4 plasmon pole models or contour integration ; non-self-consistent / partly selfconsistent / static fully self consistent ; spin-polarized) Plane Waves - Pseudopotentials, Bristol March

21 ABINIT v5.3 capabilities (II) Insulators/metals - smearings : Fermi, Gaussian, Gauss-Hermite... Collinear spin / non-collinear spin / spin-orbit coupling Total energy, charge, band structure, DOS, PDOS... Forces, stresses, automatic optimisation of atomic positions and unit cell parameters (Broyden and Molecular dynamics with damping) Molecular dynamics (Verlet or Numerov), Nosé thermostat, Langevin dynamics Susceptibility matrix by sum over states (Adler-Wiser) Optical (linear + non-linear) spectra by sum over states Electric field gradients (v5.4) Symmetry analyser (database of the 230 spatial groups and the 1191 Shubnikov magnetic groups) Plane Waves - Pseudopotentials, Bristol March

22 ABINIT v5.3 capabilities (III) Density-Functional Perturbation Theory : Responses to atomic displacements, to static homogeneous electric field, to strain perturbations Second-order derivatives of the energy, giving access to : dynamical matrices at any q, phonon frequencies, force constants ; phonon DOS, thermodynamic properties (quasi-harmonic approximation) ; dielectric tensor, Born effective charges ; elastic constants, internal strain ; piezoelectric tensor... Matrix elements, giving access to : electron-phonon coupling, deformation potentials, superconductivity Non-linear responses thanks to the 2n+1 theorem - at present : non-linear dielectric susceptibility; Raman cross-section ; electro-optic tensor Plane Waves - Pseudopotentials, Bristol March

23 ABINIT : the pipeline and the driver Parser Checks, prediction of memory needs... Summary of results CPU/Wall clock time analysis DRIVER Processing units Density, forces, MD, TDDFT... Linear Responses to atomic displacements, electric field, strains Non-linear responses Many-body perturbation theory (GW) Treatment of each dataset in turn Plane Waves - Pseudopotentials, Bristol March

24 External files in a ABINIT run Filenames Main input Pseudopotentials (previous results) ABINIT «log» Main output (other results) Results : density (_DEN), potential (_POT), wavefunctions (_WFK),... Plane Waves - Pseudopotentials, Bristol March

25 Examples (I) Charge density of graphite Plane Waves - Pseudopotentials, Bristol March

26 Examples (II) Static self-consistent GW for Cu2O, followed by excitonic calculation based on EXC (From F. Bruneval et al, Phys. Rev. Lett. 97, (2006)) TDDFT Absorption spectra of fully hydrogenated nanodiamonds up to 1 nm in diameter. (From J.-Y. Raty and G. Galli, J. Electroanalytical Chemistry 584, 9 (2005)) Plane Waves - Pseudopotentials, Bristol March

27 Examples (III) Linear response... Phonon band structure of diamond From X. G., G.-M. Rignanese and R. Caracas. Zeit. Kristall. 220, (2005) Non-linear response... Raman spectrum of MgSiO3 under pressure (Courtesy R. Caracas) Plane Waves - Pseudopotentials, Bristol March

28 Parallelism in ABINIT MPI-parallelism of the DFT part of ABINIT : Over k points : very efficient ; for metals, can lead to large speed-ups (depends on the number of k points!) ; no memory gain Over spins (if collinear spin-polarization) : very efficient ; only a factor of 2; no memory gain ; combined with k point parallelisation Over bands for linear-response calculations : efficient ; can lead to large speed-ups ; no memory gain ; combined with k point and spin parallelization Over perturbations for linear-response calculations : very efficient, limited by the number of perturbation Combined band and FFT (real space/reciprocal space) parallelism : can use distributed memory, need excellent interconnect (myrinet / infiniband), can be very good (speed up 300), still in tuning stage Combined band / FFT / k point / spin parallelism : preliminary results have shown 90% efficiency of a run with 1024 processors compared to the run with 128 processors. MPI-parallelism of the GW part of ABINIT : k-point parallelism : very efficient ; can lead to large speed-ups ; no memory gain band parallelism : very efficient ; can lead to large speed-ups ; memory gain Plane Waves - Pseudopotentials, Bristol March

29 Massive Parallelism in ABINIT : speedup A sample of gold, with a vacancy 107 atoms in a supercell 648 states (=1296 valence electrons) FFT grid 108x108x108! 1 (! r,t),! 2 (! r,t),...! N (! r,t) Bi-opterons + Infiniband Parallelisation on both states and FFT 2D-slices From François Bottin, CEA Bruyeres-le-châtel November 10, Plane Waves - Pseudopotentials, Bristol March

30 Outline I. Formalism II. III. (the PW basis set, Brillouin Zone integration, pseudopotentials, computing the forces, the PAW method) Iterative techniques (the Kohn-Sham equation, the SCF convergence, optimization of the geometry) Applications (the growth of Carbon nanotubes, the index of refraction of crystal ) IV. The ABINIT software (licence, structure, capabilities) Plane Waves - Pseudopotentials, Bristol March

31 Algorithmics : problems to be solved (1) Kohn - Sham equation #! 1 & 2 "2 + V KS (r) $ % ' ( ) i(r) = * i ) i (r) { } { r j } G j A x i =! i x i Size of the system [2 atoms 600 atoms ] + vacuum? Dimension of the vectors (if planewaves) # of (occupied) eigenvectors x i (2) Self-consistency V KS (r)! i (r) n(r) (3) Geometry optimization Find the positions { R! } of ions such that the forces { F! } vanish [ = Minimization of energy ] Current practice : iterative approaches Plane Waves - Pseudopotentials, Bristol March

32 Analysis of simple iterative algorithms

33 The steepest-descent algorithm (I) Forces are gradients of the energy : moving the atoms along gradients is the steepest descent of the energy surface. => Iterative algorithm. Choose a starting geometry, then a parameter geometry, following the forces :!, and iterately update the (n R +1) (n) (n)!," = R!," + #F!," Equivalent to the simple mixing algorithm of SCF Plane Waves - Pseudopotentials, Bristol March

34 V KS (r)! i (r) n(r) Analysis of self-consistency Natural iterative methodology (KS : in => out) : Which quantity plays the role of a force, that should vanish at the solution? The difference V in (r)! " i (r)! n(r)! V out (r) V out (r)! V in (r) (generic name : a "residual") Simple mixing algorithm ( steepest - descent ) (n v +1) in = v (n) in +! ( v (n) (n) out " v ) in Plane Waves - Pseudopotentials, Bristol March

35 Energy and forces close to the equilibrium geometry * Let us denote the equilibrium geometry as R!," Analysis of forces close to the equilibrium geometry, at which forces vanish, thanks to a Taylor expansion : * F!," (R! ',"' ) = F!," (R! ',"' ) + $ #F!," #R! ',"'! ',"' { R * } ( * R! ',"' % R ) *! ',"' + O R! ',"' % R! ',"' ( ) 2 Moreover, F!," = # $EBO $R!,"!F " ',#'!R ",# = $! 2 E BO!R ",#!R " ',#' Vector and matrix notation * R!," # R * R!," # R F!," # F! 2 E BO!R ",#!R " ',#' * { R ",# } $ H (the Hessian) Plane Waves - Pseudopotentials, Bristol March

36 The steepest-descent algorithm (II) (n R +1) (n) (n)!," = R!," + #F!," Analysis of this algorithm, in the linear regime : F(R) = F(R * )! H R! R * ( ) + O ( R! ) 2 R* R (n +1) = R (n) +!F (n) ( R (n +1)! R * ) = ( R(n)! R * )! "H ( R(n)! R * ) ( R (n +1)! R * ) = ( 1! "H)( R (n)! R * ) For convergence of the iterative procedure, the "distance" between the trial geometry and the equilibrium geometry must decrease. 1) Can we predict conditions for convergence? 2) Can we make convergence faster? Need to understand the action of the matrix (or operator) 1! "H Plane Waves - Pseudopotentials, Bristol March

37 The steepest-descent algorithm (III) What are the eigenvectors and eigenvalues of? H symmetric, positive definite matrix & = H f i = h i f i where f i The coefficient of is multiplied by 1- λ h i Iteratively : ( R (n)! R * ) = c (0) " i ( 1! #h i ) (n) f i i The size of the discrepancy decreases if 1- λ h i < 1 Is it possible to have 1- λ h i < 1, for all eigenvalues? positive definite => all h i are positive H $ & %! 2 E BO!R ",#!R " ',#' * R ",# { } { } form a complete, orthonormal, basis set ( ) = c i Thus the discrepancy can be decomposed as R (n)! R * H and ( R (n +1)! R * ) = 1! "H f i ( ) c i (n) (n) # f i = # c i ( 1! "h i )f i i i (n) " f i i Yes! If λ positive, sufficiently small... ' ) ) ( Plane Waves - Pseudopotentials, Bristol March

38 The steepest-descent algorithm (IV) ( R (n)! R * ) = c (0) " i 1! #h i i ( ) (n) f i How to determine the optimal value of λ? The maximum of all 1- λ h i should be as small as possible. At the optimal value of λ, what will be the convergence rate µ? ( = by which factor is reduced the worst component of R (n)! R *? ) ( ) As an exercise : suppose h 1 = 0.2 h 2 = 1.0 h 3 = 5.0 => what is the best value of λ? + what is the convergence rate µ? Hint : draw the three functions 1- λ h i as a function of λ. Then, find the location of λ where the largest of the three curves is the smallest. Find the coordinates of this point. Plane Waves - Pseudopotentials, Bristol March

39 The steepest-descent algorithm (V) Minimise the maximum of 1- λ h i h 1 = λ.0.2 optimum => λ = 5 h 2 = λ. 1 optimum => λ = 1 h 3 = λ. 5 optimum => λ = optimum µ = 1- λ 0.2 = 1- λ 5 h 3 h 2 h 1 λ positive? negative 1- λ. 0.2 = -( 1- λ.5) 2 - λ (0.2+5)=0 => λ = 2/5.2 µ = 1 2. (0.2 / 5.2) Only ~ 8% decrease of the error, per iteration! Hundreds of iterations will be needed to reach a reduction of the error by 1000 or more. Note : the second eigenvalue does not play any role. The convergence is limited by the extremal eigenvalues : if the parameter is too large, the smallest eigenvalue will cause divergence, but for that small parameter, the largest eigenvalue lead to slow decrease of the error... Plane Waves - Pseudopotentials, Bristol March

40 The condition number In general, λ opt = 2 / (h min + h max ) µ opt = 2 / [1+ (h max /h min )] - 1 = [(h max /h min ) -1] / [(h max /h min ) +1] Perfect if h max = h min. Bad if h max >> h min. h max /h min called the "condition" number. A problem is "ill-conditioned" if the condition number is large. It does not depend on the intermediate eigenvalues. Suppose we start from a configuration with forces on the order of 1 Ha/Bohr, and we want to reach the target 1e-4 Ha/Bohr. The mixing parameter is optimal. How many iterations are needed? Let us work this out for a generic decrease factor Δ, with "n" the number of iterations. # F (n )! h max h min "1 & % ( $ h max h min +1 ' ) # n! ln h max h min +1 &, + % (. * $ h max h min "1 '- n "1 $ F (0)! " h max h min #1 ' & ) % h max h min +1 ( ln /! 0.5 ( h max h min )ln 1 / (The latter approximate equality suppose a large condition number) n Plane Waves - Pseudopotentials, Bristol March

41 How to do a better job? 1) Taking advantage of the history Simple mixing Steepest descent very primitive algorithms Power + shift v (n) (n +1)! v The information about previous iterations is completely ignored We already know several vector/residual pairs ( v (n), r (n ) ) We should try to use them! $ R! F v in! v out " v in! T " # =! T H ˆ &! T % R = '& The residual vector is the nul vector at convergence in all three cases. 2) Decreasing the condition number ( H ˆ - # )! T Plane Waves - Pseudopotentials, Bristol March

42 Suppose we know that Minimisation of the residual (I) We try to find the best v that can be obtained by combining the latest one with its differences with other v (p). This is equivalent to What is the residual associated with v, if we make a linear approximation? # r = H( v-v * n # ) = H! s p v (p) n & & % " %! s p ( $ p=1 $ p=1 ' v* ( ' # n & n n = " s ( p v (p)! v * ) % ( = " s p H( v (p)! v * ) = " s p r (p) $ p=1 ' p=1 p=1 n-1 " v (p) v = v (n) + s p v (p)! v (n) p=1 ( ) n gives for p=1, 2... n v =! s p v (p) with 1 = s p p=1 The new residual is a linear combination of the old ones, with the same coefficients as those of the potential. An excellent strategy is select the s p such as to minimize the norm of the residual (RMM =residual minimisation method - Pulay). Then one mixes part of the predicted residual to the predicted vector. Plane Waves - Pseudopotentials, Bristol March r (p) n! p=1

43 Minimisation of the residual (II) Characteristics of the RMM method : (1) it takes advantage of the whole history (2) it makes a linear hypothesis (3) one needs to store all previous vectors and residuals (4) it does not modify the condition number Point (3) : memory problem if all wavefunctions are concerned, and the basis set is large (plane waves, or discretized grids). Might sometimes also be a problem for potential-residual pairs, represented on grids, especially for a large number of iterations. No problem of memory for geometries and forces. Simplified RMM method : Anderson's method, where only two previous pairs are kept. (D.G. Anderson, J. Assoc. Comput. Mach. 12, 547 (1964)) Plane Waves - Pseudopotentials, Bristol March

44 Modifying the condition number (I) Back to the optimization of geometry, with the linearized relation between forces, hessian and nuclei configuration : Steepest-descent : F(R) = -H( R - R * ) giving ( R (n +1)! R * ) = ( 1! "H)( R (n)! R * ) ( ) ( ) approx on the forces, and moving the nuclei approx Now, suppose an approximate inverse Hessian H -1 Then, applying H -1 R (n +1) = R (n) +!F (n) along these modified forces gives R (n +1) = R (n) +! ( H -1 ) F (n) approx The difference between trial configuration and equilibrium configuration, in the linear approximation, behaves like ( R (n +1) - R * ) = " 1 -! ( H -1 ) % $ H ' ( R (n) - R * ) # approx & Plane Waves - Pseudopotentials, Bristol March

45 F(R) = -H( R - R * ) Modifying the condition number (II) R (n +1) = R (n) +! ( H -1 ) F (n) approx ( R (n +1) - R * ) = " $ 1 -! H -1 # ( ) approx H % ' R (n) - R * & ( ) Notes : 1) If the approximate inverse Hessian is perfect, the optimal geometry is reached in one step, with λ =1. Thus the steepest-descent is NOT the best direction in general. 2) Non-linear effects are not taken into account. For geometry optimization, they might be quite large. Even with the perfect hessian, one needs 5-6 steps to optimize a water molecule. 3) Approximating the inverse hessian by a multiple of the unit matrix is equivalent to changing the λ value. 4) Eigenvalues and eigenvectors of ( H -1 ) H approx will govern the convergence : the condition number can be changed. ( H -1 ) approx is often called a "pre-conditioner". 5) Generalisation to other optimization problems is trivial. (The Hessian is referred to as the Jacobian if it is not symmetric.) Plane Waves - Pseudopotentials, Bristol March

46 Modifying the condition number (III) The approximate Hessian can be generated on a case-by-case basis. Selfconsistent determination of the Kohn-Sham potential : The Hessian matrix is the dielectric matrix. Lowest eigenvalue close to 1 usually. Largest eigenvalue for small close-shell molecules, and small unit cell solids, is (Simple mixing will sometimes converge with parameter set to 1!) For larger close-shell molecules and large unit cell insulators, the largest eigenvalue is on the order of the macroscopic dielectric constant (e.g. 12 for silicon). For large-unit cell metals, or open-shell molecules the largest eigenvalue might diverge! Model dielectric matrices are known for rather homogeneous systems. With knowledge of the approximate macroscopic dielectric constant, preconditioners can be made very efficient. Work is still in progress for inhomogeneous systems (e.g. metals/vacuum systems). Plane Waves - Pseudopotentials, Bristol March

47 Modifying the condition number (V) The approximate Hessian can be further improved by using the history. Large class of methods : - Broyden (quasi-newton-type), - Davidson, - conjugate gradients, - Lanczos... (although the three latter methods are not often presented in this way!). Plane Waves - Pseudopotentials, Bristol March

48 In ABINIT, in practice... One needs to select an history algorithm (for SCF, default is Pulay ; for geometry optimization, no default ; for Schrödinger equation, no other choice than conjugate-gradients) One needs to select a pre-conditioner (for SCF, default is Kerker, if non-metallic, it is better to provide the macroscopic dielectric constant ; for geometry optimization, no preconditioner ; for Schrödinger equation, no other choice than inverse kinetic operator) One needs to provide stopping criteria!! Also, a maximum number of iterations... (again, for Schrödinger equation, these parameters are not visible) Plane Waves - Pseudopotentials, Bristol March

49 ABINIT tutorials ABINIT Web site : http: // Tutorial page : http: // 19 lessons (from 30 minutes to 2h work)... 4 basic lessons, then independent lessons on : GW (2), phonons/dielectric constant (2), PAW (2), spin, TDDFT, optics, elastic constants, electron-phonon, nonlinear properties, parallelism,... The goal of the hands-on in Bristol : a glimpse on these lessons (only lessons 1 and 3). Convergence studies brought to a minimum!! (see lesson 2) Plane Waves - Pseudopotentials, Bristol March