Basics of Density Functional Theory (DFT)

Transcription

1 Basics of Density Functional Theory (DFT) Ari Paavo SEITSONEN Département de Chimie École Normale Supérieure, Paris École de Sidi-BelAbbès de Nanomateriaux // Octobre 8-12, 2016 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 1 / 94

2 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 2 / 94

3 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 3 / 94

4 Summary 1 Density functional theory Motivation Kohn-Sham method Exchange-correlation functional 2 Bloch theorem / supercells 3 Localised basis sets 4 Plane wave basis set 5 Pseudo potentials apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 4 / 94

5 Motivation: Why use DFT? Explicit inclusion of electronic structure Predictable accuracy (unlike fitted/empirical approaches) Knowledge of the electron structure can be used for the analysis; many observables can be obtained directly Preferable scaling compared to many quantum chemistry methods apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 5 / 94

6 History of DFT I There were already methods in the early 20th century Thomas-Fermi-method Hartree-Fock-method apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 6 / 94

7 History of DFT II Walter Kohn apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 7 / 94

8 History of DFT III: Foundations apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 8 / 94

9 Hohenberg-Kohn theorems: Theorem I Given a potential, one obtains the wave functions via Schrödinger equation: V (r) ψ i (r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 9 / 94

10 Hohenberg-Kohn theorems: Theorem I Given a potential, one obtains the wave functions via Schrödinger equation: V (r) ψ i (r) The density is the probability distribution of the wave functions: n (r) = i ψ i (r) 2 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 9 / 94

11 Hohenberg-Kohn theorems: Theorem I Given a potential, one obtains the wave functions via Schrödinger equation: V (r) ψ i (r) The density is the probability distribution of the wave functions: n (r) = i ψ i (r) 2 Thus V (r) ψ i (r) n (r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 9 / 94

12 Hohenberg-Kohn theorems: Theorem I Theorem The potential, and hence also the total energy, is a unique functional of the electron density n(r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 10 / 94

13 Hohenberg-Kohn theorems: Theorem I Theorem The potential, and hence also the total energy, is a unique functional of the electron density n(r) Thus V (r) ψ i (r) n (r) V (r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 10 / 94

14 Hohenberg-Kohn theorems: Theorem I Theorem The potential, and hence also the total energy, is a unique functional of the electron density n(r) Thus V (r) ψ i (r) n (r) V (r) The electron density can be used to determine all properties of a system apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 10 / 94

15 Hohenberg-Kohn theorems: Theorem II Theorem The total energy is variational: In the ground state the total energy is minimised apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 11 / 94

16 Hohenberg-Kohn theorems: Theorem II Theorem The total energy is variational: In the ground state the total energy is minimised Thus E[n] E[n GS ] apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 11 / 94

17 History of DFT IV: Foundations apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 12 / 94

18 History of DFT V: The reward... in 1998: apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 13 / 94

19 Kohn-Sham method: Total energy Let us write the total energy as: E tot [n] = E kin [n]+e ext [n]+e H [n]+e xc [n] E kin [n] = QM kinetic energy of electrons E ext [n] = energy due to external potential (usually ions) E H [n] = classical Hartree repulsion (e e ) E xc [n] = exchange-correlation energy apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 14 / 94

20 Kohn-Sham method: Noninteracting electrons To solve the many-body Schrödinger equation as such is an unformidable task apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 15 / 94

21 Kohn-Sham method: Noninteracting electrons To solve the many-body Schrödinger equation as such is an unformidable task Let us write the many-body wave function as a determinant of single-particle equations Then kinetic energy of electrons becomes E kin,s = i 1 2 f i ψi (r) 2 ψ i (r) f i = occupation of orbital i (with spin-degeneracy included) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 15 / 94

22 Kohn-Sham method: External energy Energy due to external potential; usually V ext = I E ext = r n (r) V ext (r) dr n (r) = f i ψ i (r) 2 i Z I r R I apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 16 / 94

23 Kohn-Sham method: Hartree energy Classical electron-electron repulsion E H = 1 n (r) n (r ) 2 r r r r dr dr = 1 n (r) V H (r) dr 2 r n (r ) V H (r) = r r r dr apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 17 / 94

24 Kohn-Sham method: Exchange-correlation energy The remaining component: Many-body complications combined = Will be discussed later apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 18 / 94

25 Total energy expression Kohn-Sham (total 1 ) energy: E KS [n] = 1 2 f i ψi 2 ψ i + n (r) V ext (r) dr i r + 1 n (r) n (r ) 2 r r r r dr dr + E xc 1 without ion-ion interaction apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 19 / 94

26 Kohn-Sham equations Vary the Kohn-Sham energy E KS with respect to ψ j (r ): δe KS δψ j (r ) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 20 / 94

27 Kohn-Sham equations Vary the Kohn-Sham energy E KS with respect to ψ j (r ): δe KS δψ j (r ) Kohn-Sham equations { 1 } V KS (r) ψ i (r) = ε i ψ i (r) n (r) = i f i ψ i (r) 2 V KS (r) = V ext (r) + V H (r) + V xc (r) V xc (r) = δexc δn(r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 20 / 94

28 Kohn-Sham equations: Notes { 1 } V KS (r) ψ i (r) = ε i ψ i (r) ; n (r) = i f i ψ i (r) 2 Equation looking like Schrödinger equation The Kohn-Sham potential, however, depends on density The equations are coupled and highly non-linear Self-consistent solution required ε i and ψ i are in principle only help variables (only ε HOMO has a meaning) The potential V KS is local The scheme is in principle exact apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 21 / 94

29 Kohn-Sham equations: Self-consistency 1 Generate a starting density n init 2 Generate the Kohn-Sham potential V init KS 3 Solve the Kohn-Sham equations ψ init i 4 New density n 1 5 Kohn-Sham potential V 1 KS 6 Kohn-Sham orbitals ψ 1 i 7 Density n apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 22 / 94

30 Kohn-Sham equations: Self-consistency 1 Generate a starting density n init 2 Generate the Kohn-Sham potential V init KS 3 Solve the Kohn-Sham equations ψ init i 4 New density n 1 5 Kohn-Sham potential V 1 KS 6 Kohn-Sham orbitals ψ 1 i 7 Density n until self-consistency is achieved (to required precision) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 22 / 94

31 Kohn-Sham equations: Self-consistency Usually the density coming out from the wave functions is mixed with the previous ones, in order to improve convergence In metals fractional occupations numbers are necessary The required accuracy in self-consistency depends on the observable and the expected apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 23 / 94

32 Kohn-Sham energy: Alternative expression Take the Kohn-Sham equation, multiply from the left with f i ψi and integrate: 1 2 f i ψ i (r) 2 ψ i (r) dr + f i V KS (r) ψ i (r) 2 dr = f i ε i r Sum over i and substitute into the expression for Kohn-Sham energy: E KS [n] = f i ε i E H + E xc n (r) V xc dr i r r apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 24 / 94

33 Exchange-correlation functional The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: Local density approximation, LDA Generalised gradient approximations, GGA Hybrid functionals... apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 25 / 94

34 Kohn-Sham method: Exchange-correlation energy The remaining component: Many-body complications combined apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 26 / 94

35 Total energy expression Kohn-Sham (total 2 ) energy: E KS [n] = 1 2 f i ψi 2 ψ i + n (r) V ext (r) dr i r + 1 n (r) n (r ) 2 r r r r dr dr + E xc 2 without ion-ion interaction apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 27 / 94

36 Kohn-Sham equations Vary the Kohn-Sham energy E KS with respect to ψ j (r ): δe KS δψ j (r ) Kohn-Sham equations { 1 } V KS (r) ψ i (r) = ε i ψ i (r) n (r) = i f i ψ i (r) 2 V KS (r) = V ext (r) + V H (r) + V xc (r) V xc (r) = δexc δn(r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 28 / 94

37 Exchange-correlation functional The Kohn-Sham scheme is in principle exact however, the exchange-correlation energy functional is not known explicitly Exchange is known exactly, however its evaluation is usually very time-consuming, and often the accuracy is not improved over simpler approximations (due to error-cancellations in the latter group) Many exact properties, like high/low-density limits, scaling rules etc are known Famous approximations: Local density approximation, LDA Generalised gradient approximations, GGA Hybrid functionals... apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 29 / 94

38 Exchange-correlation functional: LDA Local density approximation: Use the exchange-correlation energy functional for homogeneous electron gas at each point of space: E xc r n (r) eheg xc [n(r)]dr Works surprisingly well even in inhomogeneous electron systems, thanks fulfillment of certain sum rules Energy differences over-bound: Cohesion, dissociation, adsorption energies Lattice constants somewhat (1-3 %) too small, bulk modulii too large Asymptotic potential decays exponentially outside charge distribution, leads to too weak binding energies for the electrons, eg. ionisation potentials Contains self-interaction in single-particle case apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 30 / 94

39 Exchange-correlation functional: GGA Generalised gradient approximation: The gradient expansion of the exchange-correlation energy does not improve results; sometimes leads to divergencies Thus a more general approach is taken, and there is room for several forms of GGA: E xc r n (r) e xc[n(r), n(r) 2 ]dr Works reasonably well, again fulfilling certain sum rules Energy differences are improved Lattice constants somewhat, 1-3 % too large, bulk modulii too small Contains self-interaction in single-particle case Again exponential asymptotic decay in potential Negative ions normally not bound Usually the best compromise between speed and accuracy in large systems apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 31 / 94

40 Exchange-correlation functional: Hybrid functionals Hybrid functionals Include partially the exact (Hartree-Fock) exchange: E xc αe HF + (1 α)ex GGA + Ec GGA ; again many variants Works in general well Energy differences are still improved Vibrational properties slightly Improved magnetic moments in some systems Partial improvement in asymptotic form Usually the best accuracy if the computation burden can be handled Calculations for crystals appearing apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 32 / 94

41 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

42 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

43 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

44 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

45 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

46 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

47 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

48 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

49 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

50 Exchange-correlation functional: Observations The accuracy can not be systematically improved! van der Waals interactions still a problem (tailored approximations in sight) Most widely used parametrisations: GGA Perdew-Burke-Ernzerhof, PBE (1996); among phycisists Beck Lee-Yang-Parr, BLYP (early 1990 s); among chemists Hybrid functionals PBE0; among phycisists B3LYP; among chemists van der Waals included DFT-D3; empirical (pair-)potential vdw-df*; rvv10? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 33 / 94

51 Summary 1 Density functional theory 2 Bloch theorem / supercells 3 Localised basis sets 4 Plane wave basis set 5 Pseudo potentials apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 34 / 94

52 Periodic systems In realistic systems there are atoms in cubic millimetre unformidable to treat by any numerical method At this scale the systems are often repeating (crystals)... or the observable is localised and the system can be made periodic Choices: Periodic boundary conditions or isolated (saturated) cluster apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 35 / 94

53 Periodic systems apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 36 / 94

54 Periodic systems apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 36 / 94

55 Periodic systems apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 36 / 94

56 Periodic systems apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 36 / 94

57 Periodic systems Is it possible to replace the summation over translations L with a modulation? apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 37 / 94

58 Periodic systems Is it possible to replace the summation over translations L with a modulation? Bloch s theorem For a periodic potential V (r + L) = V (r) the eigenfunctions can be written in the form ψ i (r) = e ik r u ik (r), u ik (r + L) = u ik (r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 37 / 94

59 Periodic systems: Reciprocal space Reciprocal lattice vectors: b 1 = 2π a 2 a 3 a 1 a 2 a 3 b 2 = 2π a 3 a 1 a 2 a 3 a 1 b 3 = 2π a 1 a 2 a 3 a 1 a 2 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 38 / 94

60 Periodic systems: Brillouin zone First Brillouin zone: Part of space closer to the origin than to any integer multiple of the reciprocal lattice vectors, K = n 1 b 1 + n 2 b 2 + n 3 b 3 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 39 / 94

61 Integration over reciprocal space Thus the summation over infinite number of translations becomes an integral over the first Brillouin zone: L k 1.BZ In practise the integral is replaced by a weighted sum of discrete points: dk Thus eg. w k dk k k n (r) = w k f ik ψ ik (r) 2 k i apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 40 / 94

62 Periodic systems: Dispersion apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 41 / 94

63 Band structure: Example Pb/Cu(111) Photoemission vs DFT calculations for a free-standing layer Felix Baumberger, Anna Tamai, Matthias Muntwiler, Thomas Greber and Jürg Osterwalder; Surface Science (2003) doi: /s (03) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 42 / 94

64 Monkhorst-Pack algorithm Approximate the integral with an equidistance grid of k vectors with identical weight: apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 43 / 94

65 Monkhorst-Pack algorithm Approximate the integral with an equidistance grid of k vectors with identical weight: n = 2p q 1 2q, p = 1... q k ijk = n 1 b 1 + n 2 b 2 + n 3 b 3 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 43 / 94

66 Symmetry operations If the atoms are related by symmetry operation S (Sψ (r) = ψ (Sr)) the integration over the whole 1st Brillouin zone can be reduced into the irreducible Brillouin zone, IBZ apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 44 / 94

67 Symmetry operations If the atoms are related by symmetry operation S (Sψ (r) = ψ (Sr)) the integration over the whole 1st Brillouin zone can be reduced into the irreducible Brillouin zone, IBZ Sψ ik (r) = ψ ik (Sr) = e ik Sr u ik (Sr) = e ik r u ik (r), k = S 1 k apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 44 / 94

68 Symmetry operations If the atoms are related by symmetry operation S (Sψ (r) = ψ (Sr)) the integration over the whole 1st Brillouin zone can be reduced into the irreducible Brillouin zone, IBZ Sψ ik (r) = ψ ik (Sr) = e ik Sr u ik (Sr) = e ik r u ik (r), k = S 1 k dk w k = k k BZ k IBZ S w Sk apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 44 / 94

69 Irreducible Brillouin zone: Examples apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 45 / 94

70 Doubling the unit cell apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 46 / 94

71 Doubling the unit cell (super-cells) If one doubles the unit cell in one direction, it is enough to take only half of the k points in the corresponding direction in the reciprocal space apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 47 / 94

72 Doubling the unit cell (super-cells) If one doubles the unit cell in one direction, it is enough to take only half of the k points in the corresponding direction in the reciprocal space And has to be careful when comparing energies in cells with different size unless either equivalent sampling of k points is used apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 47 / 94

73 Doubling the unit cell (super-cells) If one doubles the unit cell in one direction, it is enough to take only half of the k points in the corresponding direction in the reciprocal space And has to be careful when comparing energies in cells with different size unless either equivalent sampling of k points is used or one is converged in the total energy in both cases apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 47 / 94

74 Summary 1 Density functional theory 2 Bloch theorem / supercells 3 Localised basis sets 4 Plane wave basis set 5 Pseudo potentials apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 48 / 94

75 Implementation of Kohn-Sham equations: Discretisation Kohn-Sham orbitals expanded in a basis set: ψ ik = α c ikαχ α Different choices of basis set: Localised: Gaussian (x j y k z l e (x/σ)2 ), Slater (x j y k z l e (x/σ) ), wavelet,... Extended: Plane waves Combination: FLAPW, GPW/GAPW,... apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 49 / 94

76 Summary 1 Density functional theory 2 Bloch theorem / supercells 3 Localised basis sets 4 Plane wave basis set Basics of plane wave basis set Operators Energy terms in plane wave basis set 5 Pseudo potentials apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 50 / 94

77 Kohn Sham method The ground state energy is obtained as the solution of a constrained minimisation of the Kohn-Sham energy: min E KS[{Φ i (r)}] {Φ} apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 51 / 94

78 Kohn Sham method The ground state energy is obtained as the solution of a constrained minimisation of the Kohn-Sham energy: min E KS[{Φ i (r)}] {Φ} Φ i (r)φ j (r)dr = δ ij apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 51 / 94

79 Expansion using a basis set For practical purposes it is necessary to expand the Kohn-Sham orbitals using a set of basis functions Basis set {ϕ α (r)} M α=1 Usually a linear expansion ψ i (r) = M c αi ϕ α (r) α=1 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 52 / 94

80 Plane waves Philosophy Assemblies of atoms are slight distortions to free electrons ϕ α (r) = 1 Ω e igα r (... = cos(g α r) + i sin(g α r)) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 53 / 94

81 Plane waves Philosophy Assemblies of atoms are slight distortions to free electrons ϕ α (r) = 1 Ω e igα r + orthogonal apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 53 / 94

82 Plane waves Philosophy Assemblies of atoms are slight distortions to free electrons ϕ α (r) = 1 Ω e igα r + orthogonal + independent of atomic positions apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 53 / 94

83 Plane waves Philosophy Assemblies of atoms are slight distortions to free electrons ϕ α (r) = 1 Ω e igα r + orthogonal + independent of atomic positions + no BSSE apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 53 / 94

84 Plane waves Philosophy Assemblies of atoms are slight distortions to free electrons ϕ α (r) = 1 Ω e igα r + orthogonal + independent of atomic positions + no BSSE ± naturally periodic apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 53 / 94

85 Plane waves Philosophy Assemblies of atoms are slight distortions to free electrons ϕ α (r) = 1 Ω e igα r + orthogonal + independent of atomic positions + no BSSE ± naturally periodic many functions needed apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 53 / 94

86 Computational box Box matrix : h = [a 1, a 2, a 3 ] Box volume : Ω = det h apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 54 / 94

87 Lattice vectors Direct lattice h = [a 1, a 2, a 3 ] Translations in direct lattice: L = i a 1 + j a 2 + k a 3 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 55 / 94

88 Lattice vectors Direct lattice h = [a 1, a 2, a 3 ] Translations in direct lattice: L = i a 1 + j a 2 + k a 3 Reciprocal lattice 2π(h t ) 1 = [b 1, b 2, b 3 ] b i a j = 2πδ ij Reciprocal lattice vectors : G = i b 1 + j b 2 + k b 3 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 55 / 94

89 Expansion of Kohn-Sham orbitals Plane wave expansion ψ ik (r) = G c ik (G)e i(k+g) r To be solved: Coefficients c ik (G) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 56 / 94

90 Expansion of Kohn-Sham orbitals Plane wave expansion ψ ik (r) = G c ik (G)e i(k+g) r To be solved: Coefficients c ik (G) Different routes: Direct optimisation of total energy Iterative diagonalisation/minimisation apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 56 / 94

91 Dependence on position Translation: φ(r) φ(r R I ) φ(r R I ) = G = G φ(g)e ig (r R I ) ( φ(g)e ig r) e ig R I apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 57 / 94

92 Dependence on position Translation: φ(r) φ(r R I ) φ(r R I ) = G = G φ(g)e ig (r R I ) ( φ(g)e ig r) e ig R I Structure Factor: S I (G) = e ig R I apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 57 / 94

93 Dependence on position Translation: φ(r) φ(r R I ) φ(r R I ) = G = G φ(g)e ig (r R I ) ( φ(g)e ig r) e ig R I Structure Factor: Derivatives: φ(r R I ) R I,s S I (G) = e ig R I = i G G s (φ(g)e ig r) S I (G) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 57 / 94

94 Plane waves: Kinetic energy Kinetic energy operator in the plane wave basis: ϕ G (r) = 1 2 (ig)2 1 Ω e ig r = 1 2 G 2 ϕ G (r) Thus the operator is diagonal in the plane wave basis set E kin (G) = 1 2 G 2 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 58 / 94

95 Cutoff: Finite basis set Choose all basis functions into the basis set that fulfill 1 2 G 2 E cut a cut-off sphere apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 59 / 94

96 Cutoff: Finite basis set Choose all basis functions into the basis set that fulfill 1 2 G 2 E cut a cut-off sphere apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 59 / 94

97 Cutoff: Finite basis set Choose all basis functions into the basis set that fulfill 1 2 G 2 E cut a cut-off sphere N PW 1 3/2 ΩE 2π2 cut [a.u.] Basis set size depends on volume of box and cutoff only apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 59 / 94

98 Cutoff: Finite basis set Choose all basis functions into the basis set that fulfill 1 2 G 2 E cut a cut-off sphere N PW 1 3/2 ΩE 2π2 cut [a.u.] Basis set size depends on volume of box and cutoff only and is variational! apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 59 / 94

99 Plane waves: Fast Fourier Transform The information contained in ψ(g) and ψ(r) are equivalent ψ(g) ψ(r) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 60 / 94

100 Plane waves: Fast Fourier Transform The information contained in ψ(g) and ψ(r) are equivalent ψ(g) ψ(r) Transform from ψ(g) to ψ(r) and back is done using fast Fourier transforms (FFT s) Along one direction the number of operations N log[n] apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 60 / 94

101 Plane waves: Fast Fourier Transform The information contained in ψ(g) and ψ(r) are equivalent ψ(g) ψ(r) Transform from ψ(g) to ψ(r) and back is done using fast Fourier transforms (FFT s) Along one direction the number of operations N log[n] 3D-transform = three subsequent 1D-transforms apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 60 / 94

102 Plane waves: Fast Fourier Transform The information contained in ψ(g) and ψ(r) are equivalent ψ(g) ψ(r) Transform from ψ(g) to ψ(r) and back is done using fast Fourier transforms (FFT s) Along one direction the number of operations N log[n] 3D-transform = three subsequent 1D-transforms Information can be handled always in the most appropriate space apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 60 / 94

103 Plane waves: Integrals Parseval s theorem Ω G A (G)B(G) = Ω A (r i )B(r i ) N i apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 61 / 94

104 Plane waves: Integrals Parseval s theorem Ω G A (G)B(G) = Ω A (r i )B(r i ) N i Proof. I = A (r)b(r)dr Ω = A (G)B(G) GG exp[ ig r] exp[ig r]dr = GG A (G)B(G) Ω δ GG = Ω G A (G)B(G) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 61 / 94

105 Plane waves: Electron density n(r) = ik w k f ik ψ ik (r) 2 = 1 w k f ik cik Ω (G)c ik(g )e i(g G ) r ik G,G n(r) = 2G max G= 2G max n(g)e ig r apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 62 / 94

106 Plane waves: Electron density n(r) = ik w k f ik ψ ik (r) 2 = 1 w k f ik cik Ω (G)c ik(g )e i(g G ) r ik G,G n(r) = 2G max G= 2G max n(g)e ig r The electron density can be expanded exactly in a plane wave basis with a cut-off four times the basis set cutoff. N PW (4E cut ) = 8N PW (E cut ) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 62 / 94

107 Plane waves: Operators The Kohn-Sham equations written in reciprocal space: { 1 } V KS (G, G ) ψ ik (G) = ε i ψ ik (G) However, it is better to do it like Car and Parrinello (1985) suggested: Always use the appropriate space (via FFT) There one needs to apply an operator on a wave function: O(G, G ( )ψ G ) = ( c G ) G O G G G Matrix representation of operators in: O(G, G ) = G O G Eg. Kinetic energy operator T G,G = G G = 1 2 G 2 δ G,G apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 63 / 94

108 Kohn Sham energy E KS = E kin + E ES + E pp + E xc E kin Kinetic energy E ES Electrostatic energy (sum of electron-electron interaction + nuclear core-electron interaction + ion-ion interaction) E pp Pseudo potential energy not included in E ES E xc Exchange correlation energy apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 64 / 94

109 Kinetic energy E kin = ik w k f ik ψ ik ψ ik = w k f ik cik (G)c ik(g ) k + G k + G ik GG = w k f ik cik (G)c ik(g ) Ω 1 2 k + G 2 δ G,G ik GG = Ω 1 w k f ik 2 k + G 2 c ik (G) 2 ik G apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 65 / 94

110 Periodic Systems Hartree-like terms are most efficiently evaluated in reciprocal space via the Poisson equation 2 V H (r) = 4πn tot (r) V H (G) = 4π n(g) G 2 V H (G) is a local operator with same cutoff as n tot apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 66 / 94

111 Electrostatic energy E ES = 1 2 n(r)n(r ) r r dr dr + I n(r)vcore(r)dr I + 1 Z I Z J 2 R I r J I J apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 67 / 94

112 Electrostatic energy E ES = 1 2 n(r)n(r ) r r dr dr + I n(r)vcore(r)dr I + 1 Z I Z J 2 R I r J I J The isolated terms do not converge; the sum only for neutral systems Gaussian charge distributions a la Ewald summation: [ nc(r) I = Z ( ) ] I r 2 ( ) R c 3 π 3/2 RI exp I Electrostatic potential due to nc: I n Vcore(r) I I = c (r ) r r dr = Z [ ] I r r R I erf RI R c I R c I apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 67 / 94

113 Electrostatic energy Electrostatic energy E ES = 2π Ω n tot (G) 2 G 2 G 0 + E ovrl E self E ovrl = Z I Z J R I r J L erfc R I r J L R c I 2 + R c J 2 I,J L E self = I 1 Z I 2 2π Sums expand over all atoms in the simulation cell, all direct lattice vectors L; the prime in the first sum indicates that I < J is imposed for L = 0. apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 68 / 94 R c I

114 Exchange-correlation energy E xc = n(r)ε xc (r)dr = Ω r G ε xc (G)n (G) ε xc (G) is not local in G space; the calculation in real space requires very accurate integration scheme. If the function ε xc (r) requires the gradients of the density, they are calculated using reciprocal space, otherwise the calculation is done in real space (for LDA and GGA; hybrid functionals are more intensive) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 69 / 94

115 Plane waves: Basic self-consistent cycle n(r) V xc [n](r) V KS (r) = V xc [n](r) + V ES (r) ψ ik (r) V KS (r)ψ ik (r) ψ ik (r) n (r) = ik w kf ik ψ ik (r) 2 FFT FFT N i FFT N i FFT N i FFT n(g) V ES (G) V ES (G) ψ ik (G) [V KS ψ ik ] (G) update ψ ik (G) ψ ik (G) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 70 / 94

116 Plane waves: Calculation of forces With the plane wave basis set one can apply the Hellmann-Feynman theorem (F I =) d dr I Ψ H KS Ψ = Ψ R I H KS Ψ apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 71 / 94

117 Plane waves: Calculation of forces With the plane wave basis set one can apply the Hellmann-Feynman theorem (F I =) d dr I Ψ H KS Ψ = Ψ R I H KS Ψ All the terms where R I appear explicitly are in reciprocal space, and are thus very simple to evaluate: R I e ig R I = ige ig R I apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 71 / 94

118 Plane waves: Summary Plane waves are delocalised, periodic basis functions apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

119 Plane waves: Summary Plane waves are delocalised, periodic basis functions Plenty of them are needed, however the operations are simple apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

120 Plane waves: Summary Plane waves are delocalised, periodic basis functions Plenty of them are needed, however the operations are simple The quality of basis set adjusted using a single parametre, the cut-off energy apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

121 Plane waves: Summary Plane waves are delocalised, periodic basis functions Plenty of them are needed, however the operations are simple The quality of basis set adjusted using a single parametre, the cut-off energy Fast Fourier-transform used to efficiently switch between real and reciprocal space apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

122 Plane waves: Summary Plane waves are delocalised, periodic basis functions Plenty of them are needed, however the operations are simple The quality of basis set adjusted using a single parametre, the cut-off energy Fast Fourier-transform used to efficiently switch between real and reciprocal space Forces and Hartree term/poisson equation are trivial apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

123 Plane waves: Summary Plane waves are delocalised, periodic basis functions Plenty of them are needed, however the operations are simple The quality of basis set adjusted using a single parametre, the cut-off energy Fast Fourier-transform used to efficiently switch between real and reciprocal space Forces and Hartree term/poisson equation are trivial The system has to be neutral! Usual approach for charged states: Homogeneous neutralising background apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

124 Plane waves: Summary Plane waves are delocalised, periodic basis functions Plenty of them are needed, however the operations are simple The quality of basis set adjusted using a single parametre, the cut-off energy Fast Fourier-transform used to efficiently switch between real and reciprocal space Forces and Hartree term/poisson equation are trivial The system has to be neutral! Usual approach for charged states: Homogeneous neutralising background The energies must only be compared with the same E cut apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 72 / 94

125 Summary 1 Density functional theory 2 Bloch theorem / supercells 3 Localised basis sets 4 Plane wave basis set 5 Pseudo potentials Introduction to pseudo potentials Generation of pseudo potentials Usage of pseudo potentials apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 73 / 94

126 Why use pseudo potentials? Reduction of basis set size effective speedup of calculation Reduction of number of electrons reduces the number of degrees of freedom For example in Pt: 10 instead of 78 Unnecessary Why bother? They are inert anyway... Inclusion of relativistic effects relativistic effects can be included partially into effective potentials apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 74 / 94

127 Why pp? Estimate for number of plane waves plane wave cutoff most localized function 1s Slater type function exp[ Zr] Z: effective nuclear charge φ 1s (G) 16πZ 5/2 G 2 + Z 2 Cutoff Plane waves H 1 1 Li 4 8 C 9 27 Si Ge apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 75 / 94

128 Pseudo potential What is it? Replacement of the all-electron, Z/r problem with a Hamiltonian containing an effective potential It should reproduce the necessary physical properties of the full problem at the reference state The potential should be transferable, ie. also be accurate in different environments The construction consists of two steps of approximations Frozen core approximation Pseudisation apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 76 / 94

129 Frozen core approximation Core electrons are chemically inert Core/valence separation is sometimes not clear Core wavefunctions are from atomic reference calculation Core electrons of different atoms do not overlap apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 77 / 94

130 Remaining problems Valence wavefunctions must be orthogonal to core states nodal structures high plane wave cutoff Thus pseudo potential should produce node-less functions and include Pauli repulsion Pseudo potential replaces Hartree and XC potential due to the core electrons apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 78 / 94

131 Remaining problems Valence wavefunctions must be orthogonal to core states nodal structures high plane wave cutoff Thus pseudo potential should produce node-less functions and include Pauli repulsion Pseudo potential replaces Hartree and XC potential due to the core electrons XC functionals are not linear: approximation E XC (n c + n v ) = E XC (n c ) + E XC (n v ) This assumes that core and valence electrons do not overlap. This restriction can be overcome with the non linear core correction (NLCC) discussed later apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 78 / 94

132 Atomic pseudo potentials: Basic idea 1 Create a pseudo potential for each atomic species separately, in a reference state (usually a neutral or slightly ionised atom) 2 Use these in the Kohn-Sham equations for the valence only ( T + V (r, r ) + V H (n v ) + V XC (n v ) ) Φ v i (r) = ɛ i Φ v i (r) Goal Pseudo potential V (r, r ) has to be chosen such that the main properties of the all-electron atom are imitated as well as possible apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 79 / 94

133 Pseudisation of valence wave functions apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 80 / 94

134 Generation of pseudo potentials: General recipe 1 Atomic all electron calculation (reference state) Φ v l (r) and ɛ l. 2 Pseudize Φ v l Φ PS l 3 Calculate potential from [T + V l (r)] Φ PS l (r) = ɛ l Φ PS (r) 4 Calculate pseudo potential by unscreening of V l (r) l V PS l V PS is dependent l (r) = on V l l! (r) Otherwise V H (n PS poor ) V transferability XC (n PS ) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 81 / 94

135 Norm-conserving pseudo potentials Hamann-Schlüter-Chiang-Recipe (HSC) conditions D R Hamann, M Schlüter & C Chiang, Phys Rev Lett 43, 1494 (1979) 1 Real and pseudo valence eigenvalues agree for a chosen prototype atomic configuration: ɛ l = ˆɛ l 2 Real and pseudo atomic wave functions agree beyond a chosen core radius r c : Ψ l (r) = Φ l (r) for r r c 3 Integral from 0 to R of the real and pseudo charge densities agree for R r c (norm conservation): Φ l Φ l R = Ψ l Ψ l R for R r c, Φ Φ R = R 0 r 2 φ(r) 2 dr 4 The logarithmic derivatives of the real and pseudo wave function and their first energy derivatives agree for r r c. Points 3) and 4) are related via 1 [ 2 (rφ) 2 d d dɛ dr lnφ] R = R 0 r 2 Φ 2 dr apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 82 / 94

136 Semi-local pseudo potential Final semi-local form L max V PS (r, r ) = Vloc PS (r) + L=0 V PS L (r) Y L Y L Local pseudo potential Vloc PS Non-local pseudo potential VL PS = VL PS Vloc PS Any L quantum number can have a non-local part This form leads, however, to large computing requirements in plane wave basis sets apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 83 / 94

137 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 84 / 94

138 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 85 / 94

139 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 86 / 94

140 Kleinman Bylander form Write the pseudo potential terms as Kleinman-Bylander operator ( r, r ) = V L φ L ω L V L φ L ˆV KB PS,nl φ L is the pseudo wave function E PS,nl = ik L ω L = φ L V L φ L w k f ik ψ ik V L φ L ω L V L φ L ψ ik L Now the expression is fully non-local {f (r ) f (r)} and thus computationally efficient also in plane wave basis set! apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 87 / 94

141 Ghost states Problem: In Kleinman Bylander form, the node-less wave function is no longer the solution with the lowest energy Solution: Carefully tune the local part of the pseudo potential until the ghost states disappear How to detect ghost states: Look for following properties Deviations of the logarithmic derivatives of the energy of the KB pseudo potential from those of the respective semi-local pseudo potential or all electron potential. Comparison of the atomic bound state spectra for the semi-local and KB pseudo potentials. Ghost states below the valence states are identified by a rigorous criteria by Gonze et al apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 88 / 94

142 Ultra soft pseudo potentials and PAW method Many elements require high cutoff for plane wave calculations First row elements: O, F Transition metals: Cu, Zn f elements: Ce relax norm-conservation condition n PS (r)dr + Q(r)dr = 1 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 89 / 94

143 Ultra soft pseudo potentials and PAW method Augmentation functions Q(r) depend on environment. No full un-screening possible, Q(r) has to be recalculated for each atom and atomic position. Complicated orthogonalisation and force calculations. Allows for larger r c, reduces cutoff for all elements to about 30 Ry apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 90 / 94

144 Non-linear core correction (NLCC) For many atoms (eg. alkali atoms, transition metals) core states overlap with valence states; thus linearisation of XC energy breaks down Possible cures Add additional states to valence adds more electrons needs higher cutoff Add core charge to valence charge in XC energy non linear core correction (NLCC) S.G. Louie et al., Phys. Rev. B, (1982) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 91 / 94

145 Testing of pseudo potentials Calculation of other atomic valence configurations (occupations), comparison with all-electron results Calculation of transferability functions, logarithmic derivatives, hardness Calculation of small molecules, comparison with all electron calculations (geometry, harmonic frequencies, dipole moments) Calculation of other test systems Check of basis set convergence (cutoff requirements) apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 92 / 94

146 Convergence test for pseudo potentials in O 2 apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 93 / 94

147 Pseudo potentials: Summary Pseudo potential are necessary when using plane wave basis sets in order to keep the number of the basis function manageable Pseudo potentials are generated at the reference state; transferability is the quantity describing the accuracy of the properties at other conditions The mostly used scheme in plane wave calculations is the Troullier-Martins pseudo potentials in the fully non-local, Kleinman-Bylander form Once created, a pseudo potential must be tested, tested, tested!!! apsi (ENS, UPMC, CNRS) DFT Sidi-BelAbbès 94 / 94