Density Functional Theory : Implementation in ABINIT

Transcription

1 Sherbrooke 2008 Density Functional Theory : Implementation in ABINIT X. Gonze Université Catholique de Louvain, Louvain-la-neuve, Belgium Density Functional Theory - Implementation, Sherbrooke May

2 Overview Introduction A. Density Functional Theory : our starting point B. The plane wave basis set C. Iterative algorithms. Data flow. D. Brillouin zone integration E. Pseudopotentials F. Computing the forces G. ABINIT H. The band gap problem Density Functional Theory - Implementation, Sherbrooke May

3 Density Functional Theory : our starting point

4 Our starting point : DFT (I) Non-interacting electrons in the Kohn-Sham potential : #! 1 & 2 "2 + V KS (r) $ % ' ( ) i (r) = * i ) i (r) Density n(r) = V KS (r) = V ext (r) + occ "! i * (r)! i (r) [ ] n(r 1 )! dr 1 + "E xc n r 1 - r "n(r) Hartree potential To be solved self-consistently! i Exchange-correlation potential Density Functional Theory - Implementation, Sherbrooke May

5 Our starting point : DFT (II) The solution of the Kohn-Sham self-consistent system of equations is equivalent to the minimisation of E KS {! i } occ "# $ % = (! i & 1 2 '2! i + ) V ext (r)n(r) dr i occ " with n(r) =! i * (r)! i (r) under constraints of orthonormalization for the occupied orbitals. i n(r 1 )n(r 2 ) ) dr 1 dr 2 + E xc n r 1 - r 2! i! j = " ij How to represent all these quantities in a computer? Only a finite set of numbers can be stored and treated... [ ] Density Functional Theory - Implementation, Sherbrooke May

6 Example Charge density of graphite Density Functional Theory - Implementation, Sherbrooke May

7 Accuracy, typical usage. If covalent bonds, metallic bonds, ionic bonds : 2-3% for the geometry (bond lengths, cell parameters) 0.2 ev for the bonding energies (GGA) problem with the band gap Worse for weak bonding situations (Hydrogen bonding, van der Waals). Also problematic for strongly correlated systems : electronic structure. Treatment of a few hundred atoms is OK on powerful parallel computers Up to atoms is OK on a PC. Density Functional Theory - Implementation, Sherbrooke May

8 The plane wave basis set

9 Prerequisites of plane waves Plane waves e ikr have infinite spatial extent. One cannot use a finite set of planewaves for a finite system! Need periodic boundary conditions. Primitive vectors a i, primitive cell volume! 0 a 3 OK for crystalline solids But : finite systems, surfaces, defects, polymers,...? a 1 a 2 Density Functional Theory - Implementation, Sherbrooke May

10 The supercell technique Molecule Surface : treatment of a slab Point defect in a bulk solid The supercell must be sufficiently big : convergence study Density Functional Theory - Implementation, Sherbrooke May

11 Treatment of a periodic system : wavevectors For a periodic Hamiltonian : wavevector To be considered in the Brillouin Zone k (crystal momentum) Bloch s theorem! m,k (r+a i ) = e ik.a i! m,k (r) Normalization of wavefunctions? Born-von Karman supercell! m,k (r+n i a i ) =! m,k (r) supercell vectors N i a i with N=N 1 N 2 N 3! m,k (r) = ( N" ) -1/2 0 e ik.r u m,k (r) u m,k (r+a i )=u m,k (r) Density Functional Theory - Implementation, Sherbrooke May

12 Plane wave representation of wavefunctions Reciprocal space lattice : G such that e igr has the periodicity of the real space lattice u k (r) =!!u k (G) e igr! k (r) = ( N" 0 ) -1/2!u k (G) G!u k (G) = 1 " e -igr u k (r) dr (Fourier transform)!! o o # e i(k+g)r Kinetic energy of a plane wave! "2 2 # (k+g)2 2 The coefficients!u k (G) for the lowest eigenvectors decrease exponentially with the kinetic energy ( k + G) 2 2 Selection of plane waves determined by a cut-off energy E cut ( k + G) 2 2 < E cut Plane wave sphere! Discontinuous increase of the number of the plane waves N pw Density Functional Theory - Implementation, Sherbrooke May G

13 The number of plane waves as a function of the kinetic energy cut-off Density Functional Theory - Implementation, Sherbrooke May d N PW (2 E cut ) 1 2

14 Plane wave representation of the density and potential Fourier transform of a periodic function f(r)!f(g) = 1 " e -igr f(r) dr f(r) = e -igr! "! " f(g)! o r o r! G Poisson s equation!!n (G) and!v H (G) n(r') V H (r) =! dr' " # 2 V H = -4$ n(r) r r-r' Relation between Fourier coefficients: G 2!V H (G) = 4!!n(G) VH! (G) = 4! G!n(G) 2 For G 2 =0 ( G=0 ) divergence of V H ( G=0 )!n(g=0) = 1 " n(r) dr Average!! or or Density Functional Theory - Implementation, Sherbrooke May

15 Representation of the density Density associated with one eigenfunction : n nk (r) = u * nk (r) u nk (r) Computation of u * nk (r) u nk (r) k+g k+g' = " * %!!u nk (G) e -igr # $ G & ' " %!!u nk (G') e -ig'r # $ G' & ' = %!"!u * nk (G)!u nk (G')# $ e i(g'-g)r GG' Non-zero coefficients for k+g! k+g'! sphere sphere G'-G The sphere for!n(g) has a double radius Density Functional Theory - Implementation, Sherbrooke May

16 Going from the real space to the reciprocal space n(r) = "!n(g) e igr G!sphere(2) Use of the discrete Fourier transform { r i}! { G} N y! R 2 N y G G N x N x Fast transform! R 1!n(G) = Reciprocal lattice Density Functional Theory - Implementation, Sherbrooke May N ri Real lattice: original cell { }! G r i { }! n(r ) i e-igr i { r i } : algorithm Fast Fourier Transform

17 Treatment of the continuity of space Choice of a basis (e.g. Plane waves) Truncating of the basis -> finite basis ( k+g) 2 Sphere of plane waves Discontinuous increase of the number of plane waves? Smearing of!u(g) -> Progressive incorporation of new G vectors Representation of the density 2 < E cut Sphere with a double radius in the reciprocal space Going from the real space to reciprocal space Discrete Fourier transform Grid of points + Fast Fourier Transform { r i}! { G} Density Functional Theory - Implementation, Sherbrooke May

18 Iterative algorithms Data flow

19 Algorithmics : problems to be solved # (1) Kohn - Sham equation! 1 & 2 "2 + V KS (r) $ % ' ( ) i(r) = * i ) i (r) Representation of V and ψ i : real space, planewaves, localised functions.. A x i =! i x i { r j } G j 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 Density Functional Theory - Implementation, Sherbrooke May

20 V KS (r)! i (r) n(r) Analysis of self-consistency Which quantity should vanish at the solution? The difference Natural iterative methodology (KS : in => out) : 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 Analysis... v out [ v ] = v [ in out v * ] +!v out ( v in " v * )!v in H This leads to the requirement to minimize 1- λ h i where h i are eigenvalues of!v out!v in (actually, the dielectric matrix) Density Functional Theory - Implementation, Sherbrooke May

21 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 displacement electric field Non-linear responses GW computation of band structure Treatment of each dataset in turn Density Functional Theory - Implementation, Sherbrooke May

22 Stages in the main processing unit (1) gstate.f90 (2) moldyn.f90 (3) scfcv.f90 (4) vtorho.f90 (5) vtowfk.f90 (6) cgwf.f90 (7) getghc.f90 Ground-state Molecular dynamics Self-consistent field convergence From a potential (v) to a density (rho) From a potential (v) to a wavefunction at some k-point Conjugate-gradient on one wavefunction Get the application of the Hamiltonian Density Functional Theory - Implementation, Sherbrooke May

23 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),... Density Functional Theory - Implementation, Sherbrooke May

24 Brillouin zone integration

25 From discrete states to the Brillouin zone Discrete summations over states : $ Total kinetic energy T = i! i " 1 2 #2! i n(r) =! * i (r)! i (r) Density In the periodic case : summation over energy bands and integration over the Brillouin zone Total kinetic energy T = Density n(r) = & n 1 ' n! 0k! 0 k " i 1! 0k! 0 k " f (# F $ # nk ) % nk $ 1 2 &2 % nk " f (# F $ # nk )% * nk (r)% nk (r) dk dk 1 How to treat X k dk?! ok "! ok Density Functional Theory - Implementation, Sherbrooke May

26 Brillouin zone integration 1! ok # X k dk " w k X k! ok $ [ with $ w k = 1 ] {k} w k How to chose { k} and { }? {k} Special points Weights If the integrand is periodic the integrand is continuous + derivable at all orders ( C D ) { k} homogeneous grid (1D - 2D - 3D) and w k all equal Then exponential convergence, with respect to Δk OK for semiconductors/insulators where the occupation number is independent of k within a band Convergence : one ought to test several grids with different Δk Monkhorst & Pack grids (Phys. Rev. B 13, 5188 (1976)) k 1 x k 2 x k 3 points + simple cubic, FCC, BCC... Other techniques... (tetrahedron method) Density Functional Theory - Implementation, Sherbrooke May

27 Homogeneous sampling of the Brillouin zone k y (units of 2π/b) k x (units of 2π/a) Density Functional Theory - Implementation, Sherbrooke May

28 Treatment of metals (I) Behaviour of f (! F "! nk )? Discontinuity of the integrand at the Fermi level Smearing technique ε F unoccupied f = 0 occupied f = 2 (spin) First trial : generalisation of DFT to finite temperature f(! nk ) = 1 1+e (! nk "! F )/kt f goes from 0 to 2 in an energy range of width kt E(T)! E(T=0) + "T F(T) = E - TS Problem : the T needed to recover the same convergence as for semiconductors is very high ( >> 2000 K ) Density Functional Theory - Implementation, Sherbrooke May

29 Treatment of metals (II) Better technique : the goal is to obtain E(σ = 0) from a total energy expression E(σ) with modified occupation numbers, where σ is similar to a temperature E(σ) E(σ=0) + α σ 2 + O(σ 3 ) with α small or E(σ) E(σ=0) + β σ n + O(σ n+1 ) with n > 2 f nk (! nk ) = s. Spin factor % & "(t)! dt % [ with & "(t)! dt = 1 ] t=! nk #! F $ #% Gaussian smearing!!(x) = 1" e-x2 α small Gauss - Hermite smearing!!(x) = 1" ( 3 2 # x2 ) e -x2 n = 4 but occupations can be negatives... 'Cold Smearing' (Marzari) Density Functional Theory - Implementation, Sherbrooke May

30 How many k points? Smearing width? Semiconductors - Insulators # k x N atoms Metals lattice parameter converged better than 0.5 % # k x N atoms! ! Use symmetries integration in the irreducible Brillouin zone 2D Example grid 4 x 4 = 16 3 points in the irreducible Brillouin Zone Smearing : depends on the density of electronic states (DOS) at the Fermi level s-p Metal (Al, Na...) ~ 0.04 Ha d Metal (Cu, Ag...) ~ 0.01 Ha! magnetism needs small σ Density Functional Theory - Implementation, Sherbrooke May

31 Pseudopotentials

32 Simplicity of PW basis, but need for pseudopotentials The Fourier transform theory teaches us : - details in real space are described if their characteristic length is larger than the inverse of the largest wavevector norm (roughly speaking) - quality of a plane wave basis set can be systematically increased by increasing the cut-off energy Problem : huge number of PWs is required to describe localized features (core orbitals, oscillations of other orbitals close to the nucleus) Pseudopotentials (or, in general, «pseudization») to eliminate the undesirable small wavelength features (will be the subject of a later section) Density Functional Theory - Implementation, Sherbrooke May

33 Core and valence electrons (I) Core electrons occupy orbitals that are «the same» in the atomic environment or in the bonding environment It depends on the accuracy of the calculation! Separation between core and valence orbitals : the density... n(r) = occ " i N core! i * (r)! i (r) = "! * i (r)! i (r) + "! * i (r)! i (r) = n core (r) + n val (r) i#core N val i#val «Frozen core» for i!core : " i = " i atom Density Functional Theory - Implementation, Sherbrooke May

34 Small core / Large core It depends on the target accuracy of the calculation! (remark also valid for pseudopotentials, with similar cores) For some elements, the core/valence partitioning is obvious, for some others, it is not. F atom : Ti atom : ( 1s) 2 + ( 2s) 2 ( 2p) 5 IP 1keV ev ( 1s) 2 ( 2s) 2 ( 2 p) 6 ( 3s) 2 ( 3p) 6 ( 4s) 2 ( 3d) 2 ( 1s) 2 ( 2s) 2 ( 2p) 6 ( 3s) 2 ( 3p) 6 ( 4s) 2 ( 3d) 2 IP 99.2 ev 43.3eV small core large core Gd atom : small core with n=1,2,3 shells, might include 4s, 4p, and 4d in the core. 4f partially filled Density Functional Theory - Implementation, Sherbrooke May

35 Core and valence electrons (II) Separation between core and valence orbitals : the energy... E KS {! i } occ "# $ % = (! i & 1 2 '2! i + ) V ext (r)n(r) dr i n(r 1 )n(r 2 ) ) dr 1 dr 2 + E xc n r 1 - r 2 [ ] E KS N core "# {! i } $ % = )! i & 1 2 '2! i + * V ext (r)n core (r) dr + 1 n core (r 1 )n core (r 2 ) i(core 2 * dr 1 dr 2 r 1 - r 2 N val + )! i & 1 2 '2! i + * V ext (r)n val (r) dr + 1 n val (r 1 )n val (r 2 ) i(val 2 * dr 1 dr 2 r 1 - r 2 + n val (r 1 )n core (r 2 ) * dr 1 dr 2 + E xc [ n core + n val ] r 1 - r 2 One would like to have an expression for the energy of the valence electrons... Density Functional Theory - Implementation, Sherbrooke May

36 Valence electrons in a screened potential The potential of the nuclei κ is screened by the core electrons V ion,! (r) = " V ion,! (r) = " Z val,! r " R! Z! + n (r ) core,! 1 r " R! The total energy becomes # & E = E val + " E $ % core,! ' ( + 1 2! # dr 1 r - r 1 $ + " Z core,! + n core,! (r 1 ) ' & r " R! # dr 1 % r - r ) 1 ( " (!,! ')! *! ' Z val,! Z val,! ' R! ) R! ' with E val,ks N val "# {! i } $ % =! i & 1 +. ) 2 '2! i + 1 ) V ion,* (r), - / 0 n (r) val dr i(val n val (r 1 )n val (r 2 ) 1 dr 1 dr 2 + E xc n core + n val r 1 - r 2 * [ ] Non-linear XC core correction Density Functional Theory - Implementation, Sherbrooke May

37 Removing core electrons (I) From the previous construction : valence orbitals must still be orthogonal to core orbitals ( => oscillations, slope at the nucleus...) Pseudopotentials try to remove completely the core orbitals from the simulation Problem with the number of nodes This is a strong modification of the system... Pseudopotentials confine the strong changes within a «cut-off radius» Density Functional Theory - Implementation, Sherbrooke May

38 Removing core electrons (II) Going from 1 2!2 + v " # i > = $ i " # i > To 1 2!2 + v ps " # ps,i > = $ ps,i " # ps,i > Possible set of conditions (norm-conserving pseudopotentials) : Hamann D.R., Schluter M., Chiang C, Phys.Rev.Lett. 43, 1494 (1979)! i =! ps,i! i (r) =! ps,i (r) for r > r c! i (r) 2 dr =! ps,i (r) 2 dr r < r c r < r c For the lowest angular momentum channels (s + p... d...f) Density Functional Theory - Implementation, Sherbrooke May

39 Forms of pseudopotentials Must be a linear, hermitian operator General form : ( kernel ˆV ps! )(r) = " V ps (r,r')! (r')dr' Spherically symmetric! V kernel ps (r,r') = V loc (r)!(r-r') + V nloc (r,r') Non-local part V nloc (r,r') = *! Y!m!m (",# )V! (r,r ')Y!m (" ',# ') Semi-local psp V! (r,r ') = V! (r)!(r " r ') see the paper by Bachelet, Hamann and Schlüter, Phys.Rev.B 26, 4199 (1982) Separable psp V! (r,r ') =!! * (r) f!!! (r ') Kleinman L., Bylander D.M., Phys.Rev.Lett. 48, 1425 (1982) Density Functional Theory - Implementation, Sherbrooke May

40 1.0 3s Radial wave function of Si Amplitude Pseudo-wave function All-electron wave function Radial distance (atomic units) V PS (R) (Ha) p d s! Z v r Radial distance (atomic units) Density Functional Theory - Implementation, Sherbrooke May

41 Ultra-soft pseudopotentials : the idea Problem with norm-conserving pseudopotentials : the norm conservation limits the softness! When orbitals without nodes (1s, 2p, 3d, 4f) must be treated as valence, their characteristic wavelength is quite small, and the energy cut-off is large. Idea (Vanderbilt, Phys. Rev. B 41, 7892 (1990)) Suppress the norm-conservation condition : - modify the normalization, to keep correct scattering properties - introduce charge density corrections, For selected elements, can decrease the number of PW/Grid points by a factor of two or three. More difficult to implement than norm-conserving PPs. Can be obtained as a particular case of PAW construction... Density Functional Theory - Implementation, Sherbrooke May

42 Projector Augmented Waves (PAW) Idea (P. Blöchl, PRB50, (1994)). The true wave-function! and a well-behaving pseudo-wavefunction! can be linked by a linear transformation! = T! by which physical quantities like! A! can be easily calculated in the pseudo Hilbert space representation! A!, with A = T + AT. Similarly, the variational principle for the total energy [ ]!E T "! " = #T + T " gives an equivalent of the Kohn-Sham equation, for the pseudowavefunctions, and the search for the ground state can also be done in the pseudo Hilbert space. Density Functional Theory - Implementation, Sherbrooke May

43 The transformation operator The operator T has to modify the smooth pseudowavefunction in each atomic region, to give it the correct nodal structure. Identity + sum of atomic contributions! = T! Choose : Partial waves with T =1 +! S R (R=atomic site label) R! i Pseudo partial waves = basis set, solutions of the Schrödinger Eq. for the isolated atoms within some cut-off radius! i r c,r, identical to the partial waves beyond the cut-off radius Define S such as :! i = (1 + S R )! i Density Functional Theory - Implementation, Sherbrooke May

44 The projector functions Inside each atom sphere!( r! ) = $ " i ( r! ) c i! ( r! ) = i#r $ " i ( r! ) c i i#r How to compute the coefficients of this expansion? Definition of the projector functions, dual of the pseudo partial waves :! i within the sphere, with " p i =1 i Inside the atomic sphere R Combining the different definitions : T =1 + #! i "! i i ( ) p i! ( r! ) = $ " i ( r! ) i#r p i p i!! j = " ij Density Functional Theory - Implementation, Sherbrooke May

45 Representation of the wavefunctions! = T! T =1 + #! i "! i i ( ) p i Explicitely,! = with! R 1 #! + ( 1! R "! 1 ) R R = $ " i p i! i#r! R 1 $ = " i i#r p i!! represented by plane waves (might use other representations)! R 1 and! R 1 represented on a radial grid, centered on R Note : Outside of the spheres,! R 1 Inside one sphere,! =! 1 R =! 1 R % ' = $ " i & i#r p i! ( * ) Density Functional Theory - Implementation, Sherbrooke May

46 Approximations (1) Core electrons : usually treated in the frozen-core approximation, and treated separately (see later), without PW representation (same as PPs). (2) Finite PW basis set (same as PPs) (3) The partial wave expansion is truncated : only one or two partial wave(s), for each atom R, and each l,m channel "! i p i #1 i Inside one sphere,! =! 1 R is only approximately true! % ' = $ " i &! contributes inside the atomic spheres, and corrects for the missing terms due to truncation of partial wave expansion i#r p i! ( * ) Density Functional Theory - Implementation, Sherbrooke May

47 Expectation values of the set of occupied WFs (1) A =! f n " n A " n + # c c! n A # n =! f n " n T + AT " n + # c c! n A # n n N core n =1 Matrix elements, for local or semi-local operators $! A! = &! + #! 1 R " % R 1 (! R ) n ' $ ) A &! + #! 1 R " ( % R 1 (! R ) =! A! + (! 1 1 R A! R "! 1 A! 1 ) R R +! 1 R "! 1 R A! " ' ) ( N core n =1 # (part 1) R (! 1 R + (c.c) ) R $ 1! R (part 3) R#R' # (part 2) +! 1 R "! 1 R A! 1 R " Parts 2 and 3 would vanish if the partial wave expansion were complete. They are neglected in practice. Density Functional Theory - Implementation, Sherbrooke May

48 Expectation values of the set of occupied WFs (2) A =! f N $ n " n A " 1 n + " n,r A " 1 n,r # " 1 A " ' ( 1! ) c & ) + * c c! % n,r n,r n A * n ( n =! f n " n A " n +! # c n A n R Thanks to the explicit expressions for the radial parts of the wavefunction in terms of the projector functions, this gives : N c n =1 # n c $ N c, R + " D i, j # j A # i + # c c " & " n A # n R % i, j!r n!r %! # D i, j $ j A N c, R $ i + $ c n A c # ' # $ n R & i, j "R n "R where D i, j is the one-center density matrix defined as D i, j = " p i! n f n! n n p j ' ) ( ( * ) n =1 Density Functional Theory - Implementation, Sherbrooke May

49 Example : the expression for the density Sum of PW and atom-centered contributions " ( ) n(! r ) = n (! r ) + n 1 R (! r )! n 1 R (! r ) with R n ( r! ) =! f n n " * n (! r )" n (! r ) + n c n 1 R (! r ) = # D i, j! * j (! r )! i (! r ) + n c,r n 1 R ( r! ) = i, j "R # D! i, j * ( r! )! ( r! ) + j i i, j "R n c,r Compare with norm-conserving and ultra-soft PPs... Density Functional Theory - Implementation, Sherbrooke May

50 Computing the forces

51 Computing the forces (I) Born - Oppenheimer approximation one finds the electronic ground state in the potential created by the nuclei. Consider a given configuration of nuclei { R! }. Usually it is NOT the equilibrium geometry. F!," = - #E #R!," R!"! { } (principle of virtual works) Forces can be computed by finite differences. Better approach : compute the response to a perturbation { R!," } # { R!," + $%R!," } What is the energy change? Small parameter Density Functional Theory - Implementation, Sherbrooke May

52 Computing the forces (II) Case of an electronic eigenvalue Perturbation theory : Hellmann - Feynman theorem de n d! = " n (0) dĥ d! " n (0) d" n d! is not needed! Application to an atomic displacement : H! = T " + V! ext {R! }! "H! "R #,$ = "V! ext "R #,$ F #,$ = - "E n "Ĥ = - % n % n = & n(r) "V! ext (r) ' dr "R #,$ "R #,$ "R #,$ Density Functional Theory - Implementation, Sherbrooke May

53 Computing the forces (III) Generalisation to density functional theory Reminder : E[! i ] = "! i ˆT!i + # n(r) V ext (r)dr + E Hxc [n] If change of atomic positions... V ext (! r) = "! k '!V ext (! r)!r k," = + Z k ' Z k '! r!! Rk ' n! r #! Rk 2.!! r #! R k!r k," (can be generalized to pseudopotential case) = - Z k '!! 3.!! ( r # Rk ) " r # Rk de = n(r ')!V ext(r ') $ dr ' = #!R k,"!r k," $ n(r ')! r '#! Rk 3. (! r '#! R k ) " d! r ' Density Functional Theory - Implementation, Sherbrooke May

54 Computing the forces (IV) de = n(r ')!V ext (r ') $ dr ' = #!R k,"!r k," $ n(r ')! r '#! Rk 3. (! r '#! R k ) " d! r ' Forces can be computed directly from the density! No need for additional work (like solving the Kohn-Sham equation, or self-consistency) - at variance with finite-difference approach to forces. Pseudopotentials instead of Coulomb potential... additional term, involving also wavefunctions Needed for PW/PP approach If the basis set depends on the atomic positions, and is not complete... additional term (Pulay correction, or IBSC, incomplete basis set correction)... Not needed for PW/PP approach Density Functional Theory - Implementation, Sherbrooke May

55 ABINIT

56 ABINIT in brief Chronology : Precursor : the Corning PW code (commercialized by Biosym) 1997 : beginning of the ABINIT project Dec 2000 : release of ABINITv3 under the GNU General Public License (GPL) Nov 2002 : 1st int. ABINIT developer workshop - 40 participants May 2004 : 2nd int. ABINIT developer workshop Aug 2005 : Summer School Santa Barbara - 45 students + 18 instructors Dec 2006 : CECAM school Electronic excitations - 30 students Jan 2007 : 3rd int. ABINIT developer workshop As of March 2008 : about 1250 addresses in the main mailing list, and about 30 registered developers on the repository Density Functional Theory - Implementation, Sherbrooke May

57 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 Terminology : Free software=open source=libre software Density Functional Theory - Implementation, Sherbrooke May

58 Free software licences... Many types GNU General Public Licence GNU Lesser General Public License (links are possible) BSD licence X11 license, Perl license, Python license... public domain release GNU General Public License (GPL) The licences for most softwares are designed to take away your freedom to share and change it. By contrast, the GNU GPL is intended to guarantee your freedom to share and change free software - to make sure the software is free for all its users ( grants four freedoms protection of freedom «vaccination» Density Functional Theory - Implementation, Sherbrooke May

59 ABINIT v5.5 capabilities (I) Methodologies Pseudopotentials/Plane Waves + Projector Augmented Waves (for selected capabilities) Many pseudopotential types, different PAW generators + Wavelets (BIGDFT effort) Density functionals : LDA, GGA (PBE and variations, HCTH), LDA+U (or GGA+U) + some advanced functionals (exact exchange + RPA or...) LR-TDDFT for finite systems excitation energies (Casida) GW for accurate electronic eigenenergies (4 plasmon pole models or contour integration ; non-self-consistent / partly self-consistent / quasiparticle self-consistent ; spin-polarized) Density Functional Theory - Implementation, Sherbrooke May

60 ABINIT v5.5 capabilities (II) Insulators/metals - smearings : Fermi, Gaussian, Gauss-Hermite... Collinear spin / non-collinear spin / spin-orbit coupling 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 Symmetry analyser (database of the 230 spatial groups and the 1191 Shubnikov magnetic groups) Density Functional Theory - Implementation, Sherbrooke May

61 ABINIT v5.5 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 direct 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 direct 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 Density Functional Theory - Implementation, Sherbrooke May

62 The band gap problem

63 The DFT bandgap problem (I) DFT is a ground state theory no direct interpretation of Kohn-Sham eigenenergies ε i in $ &! 1 2 "2 + V ext (r) + % n(r 1 ) ' # dr 1 + V xc (r) r 1 - r ) ( * i(r) = + i * i (r) However {ε i } are similar to quasi-particle band structure : LDA / GGA results for valence bands are accurate... but NOT for the band gap E g KS =! c -! v The band gap can alternatively be obtained from total energy differences E g = E(N+1) + E(N-1) - 2 E(N) = E(N+1) - E(N) [ correct expression! ] { } - { E(N) - E(N-1) } in the limit N (where E(N) is the total energy of the N - electron system) Density Functional Theory - Implementation, Sherbrooke May

64 The DFT bandgap problem (II) For LDA & GGA, the XC potential is a continuous functional of the number of electrons # E g KS =! c -! v! i = "E "f i [Janak's theorem] = E g = E(N+1) + E(N-1) - 2E(N) N$% In general, the XC potential might be discontinuous with the number of particle e.g. xoep E g KS! E g N electrons N+1 electrons! c -! v = E g KS ΔV xc ε c - ε v Density Functional Theory - Implementation, Sherbrooke May

65 The DFT bandgap problem (III) Comparison of LDA and GW band Structures with photoemission and Inverse photoemission experiments for Silicon. From "Quasiparticle calculations in solids", by Aulbur WG, Jonsson L, Wilkins JW, in Solid State Physics 54, (2000) Density Functional Theory - Implementation, Sherbrooke May

66 E tot (Ha) Discontinuities due to the sudden change of the number of plane waves Note atomic units are used here : 1Ha = 2 Ry = ev 1Bohr=0.529 angstrom 1 Ha/Bohr 3 =21421 GPa P (10-4 Ha/Bohr 3 ) a (Bohr) Density Functional Theory - Implementation, Sherbrooke May

67 Removal of the discontinuities of the total energy E Kinetic u(g)! 0 u(g) = 0 E Kinetic Weak u(g) u(g)! 0 u(g) = 0 ( 2E cut ) 1 2 k+g ( ) 1 2 2E cut k+g (2(E cut -E cut smearing )) 1/2 Smearing of u(g) Damping of the u(g)! progressive enlargment of the G set! removal of the discontinuities Density Functional Theory - Implementation, Sherbrooke May

68 Sources and binaries Numbering of version : x.y.z e.g Major version number (change each other year) Minor version number (new capabilities, change every three months) Usually, three «active» minor versions at all time. Life of a minor version (usually one year) : Debug status number (change during the life of a minor version number) tests availability for developers production obsolete very robust robust Density Functional Theory - Implementation, Sherbrooke May