Application of Density-Functional Theory in Condensed Matter Physics, Surface Physics, Chemistry,
Engineering and Biology, Berlin, 23 July - 1 August 2001
L4
Pseudopotentials for
ab initio electronic structure calculations
Martin Fuchs
Unite PCPM, Universite Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
Key words & ideas
\Bad idea":
Plane waves & Full-potential all-electron approach
scale of orbitals
1
core:
size of 1s bohr
Z
valence: same scale, due to orthogonality
➥ number of plane waves / Z 3
Hamiltonian matrix
size / Z 6, cpu time / Z 9
➥
could do at most diamond (C)
large total energy (components)
GaAs:
Etot ' 105
~!phonon '
30
➥
eV/pair
meV
numerical precision possible but demanding
\Good news:"
Chemical bonding,. . . determined by valence electrons
core electrons matter only indirectly
removed within frozen core approximation
eect on valence electrons can be described by a potential
➥ \linearization" of core-valence interactions
orthogonalization wiggles can be eliminated
! smooth pseudo wavefunctions
! weak pseudopotential
➥ good eciency with plane waves
\pseudoization" should be independent of system
atom ! molecule ! solid
➥ transferable pseudopotentials
➥
Norm-conserving pseudopotentials , approximate electron-ion interaction potentials
. . . work accurately & eectively (almost) for all elements, because
bonding happens outside the core region of ions
norm conservation constraint leads to physical electron density & valence band energies
clever \recipes" achieve smooth potentials , fully separable potentials , . . . improved eciency!
. . . are well controlled by proper construction and testing of the pseudopotentials
From full potentials to pseudopotentials
AE
Atom - Calculate eigenstates ! solve the (radial) Schrodinger equation: T^ + V^ AE jAE
i i = iji i
if logarithmic derivatives match . . . iterate until they do,
1
d Rout ( r
=
Rout ( r) dr
match
r
)
1
d Rin ( r
Rin ( r) dr
rmatch
)
we get an eigenstate Rl with eigenvalue l .
Any potential V (r
PS
<r
match
) giving the same logarithmic
derivative outside r
gives the same eigenvalue
Pseudopotential = exact transformation of full potential
match
T^ + V^ PS jPSi i = ijPSi i
❢
r > rmatch : PSi (r) / AE
i (r)
what normalization ?
radial wavefunction
pick an energy < 0 and integrate
outward rRout ( 0) = 0 ;!Rout ( r) r< = 0 : : : rmatch
in
in
inward R ( 1) = 0 ;! R ( r) r> = 1 : : : rmatch
. . . outside classical turning point
Al 3p
pseudo
match
r
0
all−electron
0
1
2
3
4
radius (bohr)
5
Solid: logarithmic derivatives $ boundary conditions
core ! P clmRl(r)Ylm()
P
inter
! cklmjl(kr)Ylm()
d inter (r) d Rl (r ) l
dr
= dr inter
Rl(r) core l (r) core
core:
interstitial:
Match
r
r
The pseudopotential is \weak" - cancellation theorem
kinetic energy contributions nearly cancel:
rcore
Z
0
h
i
AE
AE
AE
T
+
V
i
i d i
the pseudopotential acts like1
X
PS PS
AE PS
^
^
V j i = V j i ; jcihcjV^ AE jPS i 0
V^ PS =attractive
core
core
3s R0r j3sj2d 3s
Al 3s
wavefunction u(r)
can bind valence states, but not core states
in the core region the potential and
interstitial
3s
0
0
core
0
1
2
3
4
radius (bohr)
repulsive, conned to core
. . . if there are core states with same angular momentum l
1
Philips, Kleinman, Phys Rev 116, 287 (1959) Heine, in Solid State Physics, Vol 24 (Academic, 1970)
5
Accuracy aspect: Norm-conservation
Pseudopotential must be transferable, i.e. perform correctly in dierent environments
Pocc
❀ PS ' AE eigenvalues $ band structure & one-particle energy
i
i
e
❀ electron density $ V n r] and total energy
If we impose
core
PSi (r) = AE
(
r
)
j
r
>
r
i
,
proper electron density outside core
and norm conservation
Z r core
0
➥
➥
2
jAE
(
r
)
j
d =
i
Z r core
0
2
jPS
(
r
)
j
d
i
,
PS
AE AE
hPS
j
i
=
h
i i
i ji i 1
correct \total charge inside core radius" , proper electrostatic potential for r > rcore
boundary conditions of AE and PS wavefunctions change in the same way with energy i 1
d
; j( r)j2
2
d
Z r core
@ ln( r) 2
=
j
(
r
)
j
d
i
i
core
r
@r
0
i
... over the width of the valence bands ! correct scattering properties
PS wavefunctions change similar to AE wavefunctions
separately for each valence state ! l-dependence
➥
potential (hartree)
radial wavefunction
−4
−2
0
0
0
0
80
d
p
s
1
1
20
mid bond
... λ (Ry)
2
3
radius (bohr)
full potential
−Z/r
2
3
mid bond
radius (bohr)
3s all−electron
3s pseudo
320
4
4
3p
Pseudopotential for aluminum ...
An example ...
Accuracy aspect: Frozen-core approximation
☛
core orbitals change with chemical environment too! Eect on total energy?
Total energy:
E tot n] ! E core nc] + E valencenv ] + E valence ;corenc nv ]
two step view: change valence density ! change eective potential ! change core density
➥
second order error V ce ncd
... cancels out in total energy dierences 1
R
1
von Barth, Gelatt, Phys Rev B 21, 2222 (1980).
Pseudopotential construction
Free atom: all-electron full potential ;! pseudo valence orbitals & pseudopotential
uAE
(r )
Kohn-Sham equations for full potential ! eigenstates i (r) = lr Ylm() ... central eld
AE
"
▲
▲
#
1d
l(l + 1) + V AE nAE r]
;
+
2 dr2
2r 2
2
ul (r) = l ul (r)
n (r) =
AE
AE
AE
AE
X
occ
2
fijAE
(
r
)
j
i
Relativity: Dirac ! scalar relativistic ! non-relativistic
Z
Full potential V AE nAE r] = ; + V H nAE r] +
r
. . . XC in LDA or GGA: take same as in solid etc.
V XC nAE r]
Pseudo atom ! pseudo valence orbitals i(r) = ulr(r) Ylm()
"
#
1d
l(l + 1) + V scr n r]
;
+
l
2 dr2
2r2
2
ul(r) = l ul(r)
n(r) =
. . . formally non-relativistic Schrodinger eq.
. . . dierent for each valence state ! l-dependent
➥ screened pseudopotentials Vlscr n r] = Vlion (r) + V H n r] +
X
occ
V XC n r]
fiji(r)j2
... pseudopotential construction
ul(r) . . .
nodeless radial pseudo wavefunctions ! must meet conditions ...
➀ same valence energy levels
l = l
➁ outside cuto radius rlcut orbitals match
ul(r > rcut ) = uAE
l (r)
... implies matching of logarithmic derivatives u(1r) drd u(r)
AE
norm-conserving hljli = hAE
l jl i = 1
+ constraints for good convergence
angular momentum dependent potential, regular for r ! 0
pseudo valence density, what are E H n]
E XC n]?
➂
Vlscr (r) . . .
n(r) . . .
➥
pararametrize ul(r) and invert Schrodinger eq.
➥
nal ionic pseudopotentials ! unscreening
l
(l + 1) d2=dr2 ul (r)
Vl (r) = l ; 2r2 + 2u (r)
l
scr
Vlion (r) = Vlsrc (r) ; V H n r] ; V XC n r]
▲
... dierent schemes, e.g. Troullier-Martins: ul(r < rlcut ) = rl+1ec0+c2r +:::+c12r
dnn ul (r) ; uAE (r) j cut = 0 , n=01:::4 ➆: d2 V scr (0) = 0
➀: hl jl i = 1 ➁-➅: dr
l
r
dr2 l
2
l
12
Transferability
Test: Logarithmic derivatives
Reality/ experiment
All-electron method
compromise with needed smoothness
needed accuracy O(0:1 : : : 0:01 eV)
{
{
{
{
electronic structure
cohesive properties
atomic structure, relaxation, phonons
formation enthalpies, activation
energies, ...
modications
{ separable potentials (computational)
{ core corrections (methodic)
▲ new materials ! GaN (with 3d or not), ...
▲ new XC functionals ! GGA, ...
Characteristic tests of PP at atomic level?
)
Dl(r ) = Rl1() drd Rl(r
rdiag >rcore
norm conservation ! Dl(l ) = DlAE (l )
diag
. . . in practice: over range of valence bands?
10
log derivative (arbitary scale)
Pseudopotential method
Aluminum r
diag
=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
Monitoring transferability
Test: Congurational changes (SCF)
❏
❏
total energy (\excitations")
@ 2E (fk) = @i(fk )
@fi@fj
@fj
2
x
Ga 4s 4p
x
20
0
pseudopotential
+
Ga
−20
0
Ga
0.5
occupancy 4p
1
0
−20
frozen core
Ga
hardness (eV)
frozen core
4s
1.6
pseudopotential
error (meV)
error (meV)
Ga 4s 4p
10
chemical hardness1 (response)
@E (fk) = (f )
i k
@fi
2
−10
1
ionization (Li!Li+, Na!Na+,...)
eigenvalues (Janak theorem)
E n(fk)]
20
s ! p promotion (C, Si, Ge, ...)
1.4
frozen core
relaxed core
pseudopotential
Ga
−40
0
0.5
occupancy 4p
1
x
4p
1.2
+
2
Ga 4s 4p
0
0.5
occupancy 4p
1
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).
\Hardness tests" in practice
energy error O(a few 10 meV)?
Eigenvalues match well?
preserved?
Level splitting
;
... PP not worse than frozen core
sp
Ga
frozen core
10
0 1
pd
0
−10
pseudopotential
−20
ion
0
4p
sp
20
−5
4s
−10
pd
−15
excited
1
2
occupancy 4p
Ga
error (meV)
sp
1 2
eigenvalue (eV)
error (meV)
20
2 1
Ga
−20
s
40
0
2 0
p
+
Ga
−>sp
1 2 3 4 5 6 7 8 9
configuration
frozen core
0
pd
−20
pseudopotential
+
−40
Ga Ga
−>sp
1 2 3 4 5 6 7 8 9
configuration
Nonlinear core-valence XC (nlcv XC)
total energy & electronic structure depend just on valence electron density
☛
X
v
E n ] = hijT^ + V^lPS jii +
i
Z
V PS loc (r)nv (r)d + E H nv ] + E XC nv ]
electronic core-valence interactions mimicked by pseudoptential !
✓ electrostatic part linear in nv
✘ exchange-correlation nonlinear, terms like (nc + nv )4=3 ...
dierent in GGA & LDA!
1
pseudopotential ! Zlinearized core-valence XC
X
XC
XC v
v
E = E n ] + n (r) V XC nc + nv r] d XC functional
VlPS (r) = Vlscr n r];V H nv r];V XC nv r] PP unscreening, consistent in LDA or GGA
restoring nonlinear core-valence XC 2
E XC = E XC nv + ncfg]
VlPS (r) = Vlscr n r] ; V H nv r] ; V XC nv + nc r]
1
2
Fuchs, Bockstedte, Pehlke, Scheer, Phys Rev B 57, 2134 (1998).
Louie, Froyen, Cohen, Phys Rev B 26, 1738 (1982).
Partial core density for nlcv XC
Overlap matters only around core edge . . .
➥ can smoothen full core density inside the core
\partial core corrections"
Potassium
nc(r)
; where 0 < g(r) < 1 e.g. a polynomial
; rnlcv is the core cuto radius
2
r n(r)
5
−
18e
−3
i
density (bohr )
h
nc(r) ! 1 ; g(r)(rnlc ; r)
0.5
−
1.5e
valence
true core
model core
0
0
1
0
0
2
4
core
r
=3.2
2
3
4
radius (bohr)
5
... where nonlinear core-valence XC makes a di
erence
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, Scheer, Phys Rev B 46, 10134 (1992).
... and where linearized core-valence XC is ne
Transferability tests for K:
50
☛
frozen core
−50
see Cu: 3-4 XC valence-valence interaction
pseudopotential
−100
A test calculation helps...
K total energy
−150
−200
linearized CV XC
☛
0
0.5
occupancy 4s
1
{ spin-density functional calculations
K 4s
−3
−4
✌
−5
linearized XC
−6
−7
nlcv XC needed:
{ \soft" valence shells (alkali's!)
{ extended core states (Zn, Cd, ...)
$ varying core-valence overlap
−2
eigenvalue (eV)
error (meV)
0
linearized nlcv XC mostly sucient!
{ 1st & 2nd row, As, Se, ...
{ \two shell" cases ! all transition metals,
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
... core-valence interactions
Group-III nitrides: N 2s resonant with Ga 3d
☛
all-electron
pseudopotential
calculated vs. experimental lattice constants
3
3
1
GGA
LDA
−1
➥
relative error (%)
error (%)
GaN
AlN
−2
PW91 GGA
InN
2
0
Need for nlcv XC in GGA ?
1
−1
LDA
−3
linearized CV XC
nonlinear CV XC
−5
4.4
4.6
4.8
5.0
lattice parameter (A)
satisfactory only with cation 3
4d states
Beware of frozen core approximation!
Na
Al
Si
Ge
GaAs
nlcv XC not more important in GGA than in LDA!
Plane-wave convergence { \smoothness"
Nearly free electrons & perturbed plane-waves:
✌
k
X
i
kr
(r) = e +
G
V PS (G) ei(k+G)r
(k + G)2 ; k2
for fast convergence reduce high Fourier components of PS (G) and hGjV^ PS jG0i
modern norm-conserving schemes are good already 1
... not perfect: \coreless" 2p & 3d states still somewhat hard
➥
Choose right scheme & (dare to) increase cuto radii
1st -row & 3
4
5d elements Troullier-Martins scheme (at potential for r ! 0)
Al, Si, Ga(4d), As, ... Troullier-Martins & Hamann scheme, ... perform much alike
loss in accuracy , upper bound for rlcut
{ poor scattering properties, . . . ! atomic transferability tests tell
{ articial overlap with neighbor \cores" ... total energy error E / n(r)V (r)d ,
R
N2 dimer: rlcut = 1:5 a.u., bondlength d=2 = 1:0 a.u. ! binding energy error O(0:1 eV)
... may be acceptable
Rappe, Raabe, Kaxiras, Joannopoulos, Phys Rev B 41, 1227 (1990) Troullier, Martins, Phys Rev B 43, 1993 (1991)
Lin, Qteish, Payne, Heine, Phys Rev B 47, 4174 (1993).
1
Plane-wave cuto
in practice
Z 1
2
GPW
G
2
:::
l (GPW ) =
juPS
l (G)j 2 dG ;
0
0
Z
... for s
p
d electrons
Corresponding total energy convergence decit:
E (GPW ) =
electrons
X
i
wili (GPW )
total energy error (eV/electron)
Kinetic energy of valence electrons as measure for
plane-wave cuto energy E PW = GPW 2 (Ry) :
For the free pseudo atom:
... for atom same as in real system
➥
gives useful estimate .. too high/ too low?
➥
▲
... can't tell how much errors cancel out
Perform convergence tests on your system!
0.2
Cu fcc
Diamond
GaAs
0.1
0.0
0
20
40
60
80 100 120
plane−wave cutoff energy (Ry)
typically we see converged properties
for l . 0:1 eV
How to represent the pseudopotential operator
Atom: radial & angular momentum representation
hrjV^ jr0i =
X
lml0m0
(r;r0 )
! V (r) r2
!
X
lm
X
lm
hrjrlmihrlmjV^ jr0l0m0ihr0l0m0jr0i =
Ylm()Ylm(0) =
(r;r0 )
Ylm() Vl(r) r2 Ylm(0)
(r;r0 )
V (r ) r 2
X
lml0m0
Ylm() Vll0mm0 (r
r0) Ylm(0)
. . . local potential
Coulomb, atomic, . . .
same for all l
. . . semilocal pseudopotential
l-dependent/ angularly nonlocal, radially local
Tl + Vl(r) ; ] unl(r) = 0
nodeless unl $ ground level for l (no core!)
strict norm-conservation for ground state
Solid: reciprocal space representation, need hGjV^ jG0i ! form factor ! like in atom
pseudopotential = local potential
!
V
loc
(r) (r ; r0) +
lmax X
l
X
V loc (r)
+ short-range corrections
0)
(
r
;
r
Ylm() Vl(r) r2 Ylm(0) l=0 m=;l
Vl(r) = Vl(r) ; V loc (r)
arbitrary V loc (r) but...
. . . semilocal pseudopotentials
➥
allows to save projections by local component
V loc (r) = Vlloc (r) with lloc = lmax
✓
▲
▲
l lmax see same Vl(r) as before
local potential $ scattering for l > lmax
(norm-conservation not imposed)
transferability of separable representation
typically l = 0
1
2
(3)
s
p
d
(f )
s
2
potential (hartree)
Truncation of l-sum for l > lmax natural:
ion
r > rcore : Vl(r) / ; Zr , all l
high l : repulsive + l(lr+1)
2 angular
momentum barrier
) high-l partial waves see mostly local potential
p
δVs,p
Aluminum
0
−2
−Z/r
loc
V =Vd
full potential
−4
0
1
2
3
radius (bohr)
4
Fully separable potentials
Semilocal potentials:
Z
hGjV^ljG0i /
N N N
transformation of semilocal form
r2drjl(Gr)Vl(r)jl(G0r)
ihulVl j
V^lNL = jVhulujlV
ui
matrix multiplications
l
N O(10 6:::)
3:::
jli =
size O(10 )
Separable
potentials $ factorization:
Z
Z
!
separable Kleinman-Bylander pseudopotential $
jl(Gr)l(r)r2dr
l(r)jl(G0r)r2dr
only scalar products
size N O(103:::)
nonlocal, fully separable pseudopotential
hrjV^ jr0i = hrjV^ loc + V^ NLjr0i
lmax
X
loc
0
= V (r) (r;r ) + hrjl lmiElKB hlml jr0i
lm
jVlul i
hulVl jVlul i1=2
l l
V^lNLjli = ElKB jli
KB-energy: strength of nonlocal vs. local part
hul jVl2juli1=2
El = hu j i
l l
KB
=
average
KB-cosine
juli eigenstates of semilocal & nonlocal
Hamiltonian H^ l = T^l + V^ loc + jliElKB hlj
hrjV^lNLjul i = Vl(r)ul (r) =: hrj
~l i
) KB-potentials norm-conserving !
note:
j~li = l ; T^l ; V^ loc juli could be
calculated directly from a chosen local potential
Kleinman-Bylander pseudopotentials at work
➥
✿
Price: full nonlocality ! spectral order of
states by radial nodes not guaranteed
ghost states above/below physical valence
Example: KB-pseudopotential for As
! ZB GaAs bandstructure
levels possible
5
Ghost states detectable in free atom ...
inspect logarithmic derivatives
do spectral analysis
... readily avoided by proper choice of
local & nonlocal components
n
0
Energy (eV)
☛
−5
proper LDA bands
wrong KB potential
−10
o
Vl(r) ! V (r)
Vl(r)
loc
L
Γ
X
Analysis of the spectrum of nonlocal Hamiltonians H^ l = T^l + V^
☛
☛
➥
+ jl ihlj
= ElKB gives the reference valence level l
can compare spectra for ❏ = 0 (local potential only) ! ~i(0)
❏ arbitrary (with nonlocal potential) ! i()
for any ElKB > 0 spectra ordered like 1
ElKB
0
El
KB
0
..
~1()
~1(0)
~0() = l
~0(0)
no ghost if l < ~1(0)
➥
loc
ElKB
..
~2()
0
~2(0)
KB
El
~1() = l
~1(0)
0
KB
El
~0() ✘
0
~0(0)
ghost if l > ~1(0)
for ElKB < 0 have ~0() < ~0(0) < ~1() : : :
no ghost if l < ~0(0)
ghost if l > ~0(0)
used as ghost state criteria in fhi98PP (pswatch)
✿
1
Gonze, Stumpf, Scheer, Phys Rev B 16, 8503 (1992)
▲
Higher levels o.k. too?
s-state
energy
p-state
AE SL NL AE SL NL
✘
✌
not told by ghost state criteria,
diagonalize Hl ) all bound
levels
Ghost states
. . . where they occur, how to avoid them?
☛ local potential lloc = lmax = 2 saves computing
✓ unproblematic: 1st & 2nd row, (earth-) alkali's
Seen in logarithmic derivatives . . .
all−electron
semilocal
separable
Se 4s
✌
logarithmic derivative
10
\artically:" zero denominator in ElKB (KB-cos)
Ga, Ge, As, Se, ...
vary cuto radii of local/ nonlocal components
0
−10
rd=1.7 a.u.
10
0
−10
rd=1.9 a.u.
−2
−1
can cause strong nonlocality (large jElKB j)
0
energy (hartree)
1
\intrinsically:" numerator of ElKB large
Cu: deep V3d(r) ) ElKB 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
Other forms of pseudopotentials
Motivation - an exact transformation between AE and PS wavefunctions is1
Xn
j i = j i +
AE
PS
o
PS
jRn i ; jRn i hPS
n j i
n
n
AE
PS
o
j i = 1 + T^ jPS i
AE
. . . PS operators (acting on pseudo wavefunctions) act as
O^ PS
^ T^
T^ yO
n
o
X
PS
AE
AE
PS
PS
^+
^ jRn0 i ; hRn jO
^ jRn0 i hPS
= O
ji i hRn jO
n0 j
nn0
=
! f: : :g looks like
➥
1
PS
PS
nn0 jn iVnn0 hn0 j
Can make ansatz for separable pseudopotential with multiple projectors2
hrjV^ljr0i = hrjV^ loc jr0i +
...
...
P
X
nn0=12:::
hrjiliV lnn0 hn0ljr0i
nl : e.g. atomic functions derived from j~nli = n ; T^l ; V^ loc junli
n = n0 = 1 like Kleinman-Bylander pseudopotentials
Blochl, Phys Rev B 50, 17953 (1994).
2
Blochl, Phys Rev B 41, 5414 (1990).
➁
Norm-conserving:
AE core
j
u
Qn n0 = huAE
nl n0lir ; hunljun0lircore = 0
. . . several reference states possible!
\Quasi" norm-conserving:
AE
hunljun0lircore + Qn n0 = huAE
nl jun0lircore
➞
Ultrasoft pseudopotentials
1
logarithmic derivative match as in norm-conserving case
density & wavefunction ! smooth part + augmentation
gives generalized eigenvalue problem H^ ; S^jii = 0
reduced plane-wave basis for 1st row & d-metal elements
➥
➥
➥
$
radial wavefunction R2p(r)
➀
all−electron
Oxygen 2p
norm conserving
pseudo
1
ultrasoft pseudo
0
0
1
radius (bohr)
2
increase in projections, added complexity
1
Vanderbilt, Phys Rev B 41, 7892 (1990) Laasonen et al, Phys Rev B 47, 10142 (1993) Kresse, Hafner, J Phys Cond Mat
6, 8245 (1994) Dong, Phys Rev B 57, 4304 (1998).
Summary
Pseudopotential = electron-ion 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
generalizations