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PHYSICAL REVIEW B 15 JANUARY 1999-IVOLUME 59, NUMBER 3US-PP concept by combining ideas from pseudopotential
and LAPW ~linearized augmented-plane-wave! methods in a
conceptually elegant framework, called the projector
augmented-wave method ~PAW!. Although Blo
¨
chl has indi-
cated in his work that similarities between ultrasoft PP’s and
his method exist, no formal relationship was established. In
the present work, we will derive this relationship, which
shows that the only difference between Vanderbilt’s and
Blo
¨
chl’s approach are simple one-center terms. We will also
indicate a simple way to implement the PAW method in
existing plane-wave codes that use norm-conserving or ultra-
soft pseudopotentials.
Norm-conserving pseudopotentials were first introduced
and used by Hamann, Schlu
¨
ter, and Chiang.
6
In their scheme,
inside some core radius, the all-electron ~AE! wave function
is replaced by a soft nodeless pseudo- ~PS! wave function,
with the crucial restriction that the PS wave function must
have the same norm as the all-electron wave function within
the chosen core radius; outside the core radius the PS and AE
wave function are identical. It is now well established that
good transferability requires a core radius around the outer-
most maximum of the AE wave function, because only then
localized atom-centered augmentation charges are intro-
duced. These augmentation charges are defined as the charge
density difference between the AE and the PS wave function,
but for convenience they are pseudized to allow an efficient
treatment of the augmentation charges on a regular grid. The
core radius of the pseudopotential can now be chosen around
half the nearest-neighbor distance—independent of the posi-
tion of the maximum of the AE wave function ~see Ref. 10!.
Only for the augmentation charges a small cutoff radius must
be used to restore the moments and the charge distribution of
the AE wave function accurately ~for details see Ref. 11!.
The pseudized augmentation charges are usually treated on a
regular grid in real space, which is not necessarily the same
as the one used for the representation of the wave functions.
The relation between the ultrasoft PP method and other
plane-wave-based methods was discussed in detail by
Singh.
5
Vanderbilt’s approach is now adopted quite widely,
11–19
and especially for the 3d transition-metals savings in the
computer and improvements in the accuracy can be
significant.
20
But the success of the method is partly ham-
pered by the rather difficult construction of the pseudopoten-From ultrasoft pseudopotentials to th
G. K
Institut fu
¨
r Theoretische Physik, Technische Universita
¨
t W
D. Jo
Physics Department, University of the Witwaters
~Received
The formal relationship between ultrasoft ~US! V
augmented wave ~PAW! method is derived. It is shown
can be obtained by linearization of two terms in a slig
ton operator, the forces, and the stress tensor are deriv
implement the PAW method in existing plane-wave c
addition, critical tests are presented to compare the ac
potential method with relaxed core all electro
(H
2
,H
2
O, Li
2
,N
2
,F
2
,BF
3
, SiF
4
) and several bulk s
Particular attention is paid to the bulk properties and
@S0163-1829~98!00848-0#
I. INTRODUCTION
First-principles Kohn-Sham density-functional methods
~see, e.g., Refs. 1 and 2! employing a plane-wave basis set
and the pseudopotential ~PP! approximation are currently
among the most successful techniques in computational
chemistry and computational material science.
3–5
The big-
gest merit of these methods is their formal simplicity, but
unfortunately this simplicity has a price: first-row elements,
transition metals, and rare-earth elements are computation-
ally demanding to treat with standard norm-conserving
pseudopotentials.
6
Therefore, various attempts have been
made to generate soft pseudopotentials, and the most suc-
cessful approach to date is the concept of ultrasoft PP intro-
duced by Vanderbilt.
7
Blo
¨
chl
8
has further developed thePRB 590163-1829/99/59~3!/1758~18!/$15.00projector augmented-wave method
sse
n, Wiedner Hauptstrasse 8-10/136, A-1040 Wien, Austria
bert
nd, P.O. Wits 2050, Johannesburg, South Africa
uly 1998!
derbilt-type pseudopotentials and Blo
¨
chl’s projector
at the total energy functional for US pseudopotentials
ly modified PAW total energy functional. The Hamil-
for this modified PAW functional. A simple way to
es supporting US pseudopotentials is pointed out. In
racy and efficiency of the PAW and the US pseudo-
methods. These tests include small molecules
tems ~diamond, Si, V, Li, Ca, CaF
2
, Fe, Co, Ni!.
gnetic energies of Fe, Co, and Ni.
the charge distribution and moments of the AE wave func-
tions are well reproduced by the PS wave functions ~see,
e.g., Ref. 9!. Therefore, for elements with strongly localized
orbitals ~like first-row, 3d , and rare-earth elements! the re-
sulting pseudopotentials require a large plane-wave basis set.
To work around this, compromises are often made by in-
creasing the core radius significantly beyond the outermost
maximum in the AE wave function. But this is usually not a
satisfactory solution because the transferability is always ad-
versely affected when the core radius is increased, and for
any new chemical environment, additional tests are required
to establish the reliability of such soft PP’s.
An elegant solution to this problem was proposed by
Vanderbilt.
7
In his method, the norm-conservation constraint
is relaxed and to make up for the resulting charge deficit,1758 1999 The American Physical Society

Otials, i.e., too many parameters ~several cutoff radii! must be
chosen and therefore extensive tests are required in order to
obtain an accurate and highly transferable PP.
Some of these disadvantages are avoided in Blo
¨
chl’s
PAW method. Blo
¨
chl introduces a linear transformation
from the PS to the AE wave function and derives the PAW
total energy functional in a consistent manner applying this
transformation to the KS functional. The construction of
PAW datasets is easier because the pseudization of the aug-
mentation charges is avoided, i.e., the PAW method works
directly with the full AE wave functions and AE potentials.
This is achieved using radial support grids around each atom
instead of regular grids. The decomposition into radial grid
and regular grid is complete, insofar that no cross term be-
tween the grids must be evaluated. Despite these advantages,
the method is not yet used very often, and in addition to
Blo
¨
chl’s own implementation of the method we are aware of
only one second program supporting the PAW method.
21,22
This is partly due to the fact that the PAW approach was
introduced a few years after Vanderbilt’s method, but an-
other reason is that—apart from its formal elegance—it was
not obvious at the time that the PAW method has significant
advantages over other frozen core approaches like the US-PP
approach. There are also some aspects in Blo
¨
chl’s work that
deviate so significantly from conventional pseudopotential
methods, that the implementation and testing of the method
seems to be fairly difficult. In this work, we will rewrite the
PAW total energy functional so that it resembles more
closely the usual expressions used in pseudopotential pro-
grams and we will establish the exact formal relationship
between both the US-PP and the PAW method. Our results
show that only very few additional terms must be evaluated
in order to implement the PAW method in programs support-
ing US-PP’s.
The paper is organized as follows: Section II derives the
rearranged PAW total energy functional. Then we establish
the formal relationship between the PAW and the US-PP
method ~Sec. II F!. The Hamilton operator and the forces for
the modified PAW functional are obtained in Sec. III, and in
Sec. IV the construction of our PAW datasets is discussed.
Several critical tests for dimers, small molecules, and bulk
systems ~including magnetic Fe, Co, and Ni! are presented in
Sec. V. Discussions and conclusions are at the end.
II. THE PAW TOTAL ENERGY FUNCTIONAL
A. Basic PAW formalism
As a first step, we derive a modified form of the PAW
total energy functional. We do that in order to obtain a func-
tional that resembles closely the functional for US-PP. Our
derivation follows Blo
¨
chl’s work closely,
8
but the decompo-
sition of the Hartree energy—the treatment of the core va-
lence interaction particularly—and the treatment of the ex-
change correlation differ somewhat. Although it would be
possible to start immediately from the final expression of the
PAW total energy functional in Ref. 8, we have decided to
rederive the modified PAW functional directly from the
Kohn-Sham density functional, because this makes the deri-
PRB 59 FROM ULTRASOFT PSEUDvation more concise and easier to follow. The exact Kohn-
Sham density functional is as usually given byE5
(
n
f
n
^
C
n
u2
1
2
DuC
n
&
1E
H
@n1n
Z
#1E
xc
@n#. ~1!
E
H
@n1n
Z
# is the Hartree energy of the electronic charge
density n and the point charge densities of the nuclei n
Z
,
E
xc
@n# is the electronic exchange-correlation energy, and f
n
are orbital occupation numbers. We will first give a brief
summary of the basics of the PAW method ~see Ref. 8; in
general we also adopt the notation of Ref. 8!. In the PAW
method, the AE wave function C
n
is derived from the PS
wave function C
˜
n
by means of a linear transformation:
8
uC
n
&
5uC
˜
n
&
1
(
i
~ uf
i
&
2uf
˜
i
&
)
^
p˜
i
uC
˜
n
&
. ~2!
The PS wave functions C
˜
n
are the variational quantities. The
index i is a shorthand for the atomic site R, the angular
momentum numbers L5l ,m , and an additional index k re-
ferring to the reference energy e
kl
. The AE partial waves f
i
are obtained for a reference atom, the PS partial waves f
˜
i
are
equivalent to the AE partial waves outside a core radius r
c
l
and match continuously onto f
˜
i
inside the core radius. The
core radius r
c
l
is usually chosen approximately around half
the nearest-neighbor distance.
10
The projector functions p˜
i
are dual to the partial waves:
^
p˜
i
uf
˜
j
&
5d
ij
.
Starting from Eq. ~2! it is possible to show that in the PAW
method, the AE charge density is given by ~for details we
refer to Ref. 8!:
n~r!5 n˜~r!1n
1
~r!2 n˜
1
~r!, ~3!
where n˜ is the soft pseudo-charge-density calculated directly
from the pseudo-wave-functions on a plane-wave grid @Eq.
~15! of Ref. 8#:
n˜~r!5
(
n
f
n
^
C
˜
n
ur
&^
ruC
˜
n
&
. ~4!
The onsite charge densities n
1
and n˜
1
are treated on a radial
support grid, that extends up to r
rad
around each ion. They
are defined as @Eq. ~16! of Ref. 8#
n
1
~r!5
(
~ i , j !
r
ij
^
f
i
ur
&^
ruf
j
&
, ~5!
and @Eq. ~17! of Ref. 8#
n˜
1
~r!5
(
~ i , j !
r
ij
^
f
˜
i
ur
&^
ruf
˜
j
&
. ~6!
r
ij
are the occupancies of each augmentation channel (i , j)
and they are calculated from the pseudo-wave-functions ap-
plying the projector functions:
r
ij
5
(
n
f
n
^
C
˜
n
up˜
i
&^
p˜
j
uC
˜
n
&
. ~7!
For a complete set of projectors the charge density n˜
1
is
1759POTENTIALS TO THE...exactly the same as n˜ within the augmentation spheres. In
addition, comparison with the work of Vanderbilt @Eqs.