Constructing pseudopotentials for ABINIT: about the use of the fhi98PP package.
Martin Fuchs. Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany.
Constructing pseudopotentials for ABINIT: about the use of the fhi98PP package Martin Fuchs Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany Unit´e PCPM, Universit´e Catholique de Louvain, Louvain-la-Neuve, Belgium
Email:
[email protected]
Description of fhi98PP 1 Generate norm-conserving pseudopotentials (DFT) of – Hamann or Troullier-Martins type – test their transferability on the atomic level, including the fully-separable KB form by a free fortran77 code, operated through commandline interface using UNIX csh scripts 2 View data by GNU GPL plotting tool xmgrace http://plasma-gate.weizmann.ac.il/Grace/ 3 Learn from tutorial (ps booklet) and background write-up M Fuchs, M Scheffler, Comput Phys Commun 119, 67-98 (1999) ➜ download from http://www.FHI-Berlin.MPG.DE/th/fhi98md/fhi98PP/ Interfaced to ABINIT. . .
. . . gives flexibility where needed
Part I: psatom Free atom: all-electron full potential −→ pseudo valence orbitals & pseudopotential
• Kohn-Sham equations for full potential → eigenstates φAE i (r) = "
#
2
1 d l(l + 1) AE AE AE AE AE − + + V [n ; r] u (r) = ² νl νl uνl (r), 2 dr 2 2r 2
uAE νl (r) Y (Ω) lm r
AE
n (r) =
X
... spherial AE
fi|φi (r)|
occ
▲ Relativity: Dirac → scalar relativistic → non-relativistic
Z H AE + V [n ; r] + V XC [nAE ; r] r . . . XC in LDA or GGA: take same as in solid etc.
▲ Full potential
V
AE
AE
[n ; r] = −
• Pseudo atom → pseudo valence orbitals φi(r) = "
2
#
ul (r) r Ylm (Ω)
l(l + 1) 1 d scr + + V [n; r] ul (r) = ²l ul (r), − l 2 dr 2 2r 2
and density n
n(r) =
X occ
. . . formally non-relativistic Schr¨ odinger eq. . . . different for each valence state → l-dependent H XC ➥ Ionic pseudopotentials Vlps [n; r] = V scr [n; r] l (r) − V [n; r] − V
fi|φi(r)|
2
2
Part I: psatom . . . is transferable?
Pseudopotential on screen. . . Ionic Pseudopotentials
• Start with default cutoff radii & neutral atom:
al:abinit 18:22:32 Nov 01 2002 fuchs
Aluminum - Hamann - LDA 2.0
• Norm-conservation conditions ²l = ²AE νl AE hφl|φli = hφAE νl |φνl i
0 rc=1.212
ps
V (r) (hartree)
1.0
1 rc=1.546 2 rc=1.546
0.0
➥ ensure correct scattering properties around atomic valence energies
-1.0
-2.0
-3.0
0.0
1.0
2.0 3.0 r (bohr)
4.0
5.0
• But how about in the solid or molecule? Must test! . . . and if needed improved! • Proposal of how to . . .
Part II: Monitoring transferability with pswatch Test: Logarithmic derivatives
Pseudopotential method Reality/ experiment All-electron method
Dl (r
diag
, ²) =
¯
¯ 1 d R (r, ²) ¯ diag l Rl (²) dr r
norm conservation: o.k. for ²l ± δ² . . . in practice: over range of valence bands? diag
Characteristic tests of PP at atomic level?
10
log derivative (arbitary scale)
◦ compromise with needed smoothness • needed accuracy ∼ O(0.1 . . . 0.01 eV) – electronic structure – cohesive properties – atomic structure, relaxation, phonons – formation enthalpies, activation energies, ... • modifications – separable potentials (computational) – core corrections (methodic) ▲ new materials → GaN (with 3d or not), ... ▲ new XC functionals → GGA, ...
Aluminum r
=2.9 bohr lloc=2
0 −10
d
all−electron all−electron semilocal semilocal separable separable
10 0 p
−10
10 0 −10
valence state −1.0
s 0.0
energy (hartree)
1.0
Hardness tests - a ,,good” example ❏ s → p promotion (C, Si, Ge, ...)
Test: Configurational changes (∆SCF)
❏ ionization (Li→Li+, Na→Na+,...)
• eigenvalues (Janak theorem) • chemical hardness1
• total energy (excitations)
2
Ga 4s 4p
x
20
10
frozen core
0
pseudopotential Ga
−20 0
+
Ga 0.5 occupancy 4p
1
4s
1.6
pseudopotential
−10
1
2
Ga 4s 4p
x
error (meV)
error (meV)
20
∂²i(fk ) ∂ 2E(fk ) = ∂fi∂fj ∂fj
∂E(fk ) = ²i(fk ) ∂fi
0
−20
frozen core
hardness (eV)
E[n(fk )]
4p
frozen core relaxed core pseudopotential
Ga
−40 0
0.5 occupancy 4p
1
x
1.4
1.2
+
Ga
2
Ga 4s 4p
0
0.5 occupancy 4p
Grinberg, Ramer, Rappe, Phys Rev B 63, 201102 (2001); Filipetti et al, Phys Rev B 52, 11793 (1995); Teter, Phys Rev B 48, 5031 (1993).
1
Nonlinear core-valence XC
2
• restoring nonlinear core-valence XC E XC = E XC [nc + nv ] different unscreening: ¯ ps scr H v XC c v ¯ Vl = Vl − V [n ] − V [n + n ] atom
• a smooth model core density is used, replaces the true core density inside cutoff radius r core
1
M Fuchs et al, Phys Rev B 57, 2134 (1998)
2
SG Louie et al, Phys Rev B 26, 1738 (1982)
0.5
Potassium
2
r n(r) 5 −
density (bohr )
18e
−3
• Pseudopotential mimicks also interactions of valence and core electrons ✓ electrostatic linear in nv ✗ exchange-correlation nonlinear, terms like (nc + nv )4/3 . . . • so far: linearized core-valence XC . . . different in LDA & GGA 1
−
1.5e
valence true core model core
0
0
1
0
0 core
r
2
4
=3.2
2 3 4 radius (bohr)
5
... where nonlinear core-valence XC makes a difference Rocksalt (NaCl):
1
✘ semi-metal with linearized CV XC (a) ✓ insulator with nonlinear CV XC (b)
0 nlcv LDA
binidng energy (eV)
−2
NaCl a0 = 10.66 bohr
−4
B0 = 24 GPa EB = 6.5 eV
−6
lcv LDA
−8
−10
6
7
8
9 10 11 12
lattice constant (bohr)
1
Hebenstreit, Scheffler, Phys Rev B 46, 10134 (1992).
... and where linearized core-valence XC is fine Transferability tests would tell. . . 50 frozen core
−50 pseudopotential −100
A test calculation helps...
K total energy −150 −200
linearized CV XC 0
0.5 occupancy 4s
1
−2 K 4s −3 eigenvalue (eV)
error (meV)
0
☛ linearized nlcv XC mostly sufficient! – 1st & 2nd row, As, Se, ... – “two shell” cases → all transition metals, see Cu: 3-4 XC valence-valence interaction
☛ nlcv XC needed: – “soft” valence shells (alkali’s!) – extended core states (Zn, Cd, ...) ↔ varying core-valence overlap – spin-density functional calculations
−4 −5 linearized XC
−6 −7
0
0.5 occupancy 4s
1
✌ turning semi-core into valence states? – Zn 3d, Ca 3d, Rb 4p, ... – Ga 3d, In 4d in III-nitrides (but not GaP, GaAs, ...) ... a bit system dependent
Kleinman-Bylander potentials at work ✓ fully separable potentials computational must
are
Example: KB-pseudopotential for As → ZB GaAs bandstructure
☛ full nonlocality → spectral order of states by radial nodes not guaranteed ➥ ghost states above/below valence levels possible
5
physical
✿ Ghost states detectable in free atom ...
• inspect logarithmic derivatives • do spectral analysis ... readily avoided by proper choice of local & nonlocal components n o loc Vl (r) → V (r), δVl (r)
Energy (eV)
0
−5 proper LDA bands wrong KB potential
−10
L
Γ
X
ˆ l = Tˆl + Vˆ loc + |χl iλhχl | Analysis of the spectrum of nonlocal Hamiltonians H ☛ λ = ElKB gives the reference valence level ²l ☛ can compare spectra for ❏ λ = 0 (local potential only) → ²˜i(0) ❏ λ arbitrary (with nonlocal potential) → ²i(λ) ➥ for any λ, ElKB > 0 spectra ordered like
λ ElKB 0 ElKB 0
...
λ ²˜1(λ) ²˜1(0) ²˜0(λ) = ²l ²˜0(0)
no ghost if ²l < ²˜1(0)
1
▲ Higher levels o.k. too?
...
ElKB 0 ElKB 0 ElKB 0 ghost if
²˜2(λ) ²˜2(0) ²˜1(λ) = ²l ²˜1(0) ²˜0(λ) ✘ ²˜0(0) ²l > ²˜1(0)
s-state AE SL NL
energy
p-state AE SL NL
✘
➥ for λ, ElKB < 0 have ²˜0(λ) < ²˜0(0) < ²˜1(λ) . . . no ghost if ²l < ²˜0(0)
ghost if ²l > ²˜0(0)
✿ used as ghost state criteria in fhi98PP (pswatch) X Gonze, R Stumpf, M Scheffler, Phys Rev B 16. 8503 (1992)
✌ not told by ghost state criteria, diagonalize Hl ⇒ all bound levels
. . . in case of ghost states Seen in logarithmic derivatives . . .
☛ local potential lloc = lmax = 2 saves computing
all−electron semilocal separable
Se 4s
✓ unproblematic: 1st & 2nd row, (earth-) alkali’s
10 logarithmic derivative
. . . where they occur, how to avoid them?
✌
can cause strong nonlocality (large |ElKB |)
0 −10
“artifically:” ≈ zero denominator in ElKB (KB-cos) Ga, Ge, As, Se, ... vary cutoff radii of local/ nonlocal components
rd=1.7 a.u.
10 0
“intrinsically:” numerator of ElKB large
−10
rd=1.9 a.u.
−2
−1
0
energy (hartree)
1
Cu: deep V3d (r) ⇒ ElKB À 0 to get 4s right all 3,4,5d-metals: Cu, Pd, Ag, ... make local potential repulsive → ElKB < 0 use to s - or p-component ! ➜ KB-potentials work well in practice
Overview - fhi98PP covers “classic” pseudopotentials Pseudopotential = electron-i on interaction • nucleus’ Coulomb attraction + core-valence interaction (orthogonality, electrostatic, XC) • work throughout periodic table (... almost) ✓ physically motivated approximation • Valence electrons rule chemical bonding • Frozen-core approximation (depends on system) • Cancellation of potential and kinetic energy in core ✓ well controlled • norm-conservation (built in) • nonlinear core-valence XC (depends on system) ✓ Transferability properties & pseudopotential validation • logarithmic derivatives (scattering properties), chemical hardness ◦ plane-wave convergence ✓ Fully separable, nonlocal potentials • analysis & removal of ghost states ✓ present XC: LDA & PW91,BP86,PBE,revPBE,RPBE,BLYP GGA . . . easy to add
• experimental XC: xOEP with KLI approximation, SIC, MGGA ✗ desirable: spin-polarization, L · S -coupling, desirable: potential & performance database on the Web ?
